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WORKS  OF  PROFESSOR  MERRIMAN 

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METHOD  OF  LEAST  SQUARFS      8vo,  238  pa.ces.  $2.00 
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By  Professors  MERRIMAN  and  JACOBY 

TEXT-BOOK  O.s    UOoFS  A  No   I,  RIDGES: 

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Edited  by  Professors  MERRIMAN  and  WOODWARD 

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Eleven  Volumes,  Svo.  Each,  $1.00 


ELEMENTS  ijjM   - 

PF 

PRECISE    SURVEYING 

AND 

GEODESY. 


BY 

MANSFIELD    MERRIMAN, 

PROFESSOR  OF  CIVIL  ENGINEERING  IN  LEHIGH  UNI\ 


SECOND    EDITION,  REVISED. 
SECOND   THOUSAND. 


NEW   YORK; 
JOHN   WILEY   &   SONS. 

LONDON:    CHAPMAN  &    HALL,    LIMITED. 
1907. 


v,i  /f^o^ijA.! 


Engineering 
Library 

Copyright,  1899, 

BY 
MANSFIELD   MERRIMAN. 


ROBERT  DRUMMOND,   PRINTER,   NEW  YORK. 


jl 
PREFACE. 


THIS  volume  is  designed  as  an  introduction  to  those  precise 
methods  of  surveying  which  are  coming  more  and  more  into 
use  in  America  as  land  becomes  more  valuable.  The  theory 
at  the  foundation  of  these  methods  is  that  of  the  principle  of 
least  squares,  which  shows  both  how  to  obtain  the  most  proba- 
ble value  of  a  measured  quantity  and  how  to  ascertain  its 
degree  of  precision.  Four  chapters  are  devoted  to  this  theory 
and  to  its  application  in  common  triangulations,  base-line 
measurements,  and  leveling.  Then  follows  the  consideration 
of  the  earth  as  a  sphere  and  as  a  spheroid,  with  methods  for 
determining  azimuth  and  for  the  execution  of  triangulations 
which1  require  the  figure  of  the  earth  to  be  taken  into  account. 
The  treatment  has  been  made  as  concise  and  elementary  as 
possible,  the  volume  being  designed  as  a  text-book  for  engi- 
neering students  and  a  manual  for  surveyors  rather  than  as  a 
treatise  for  geodeticians.  In  this  edition  all  known  errors 
have  been  corrected,  several  new  problems  have  been  inserted, 
and  new  matter  on  map  projections  and  on  American  geodetic 
work  has  been  added  in  Arts.  68  and  86. 

M.  M. 


793981 


CONTENTS, 


CHAPTER  I. 

THE   METHOD  OF   LEAST   SQUARES. 

ART.  PAGE 

1.  ERRORS  OF  OBSERVATIONS 7 

2.  LAW  OF  PROBABILITY  OF  ERROR .*   •.  ^ 

3.  THE  PRINCIPLE  OF  LEAST  SQUARES 12 

4.  WEIGHTED  OBSERVATIONS 14 

5.  OBSERVATION  EQUATIONS 16 

6.  INDIRECT  OBSERVATIONS  OF  EQUAL  WEIGHT 19 

7.  INDIRECT  OBSERVATIONS  OF  UNEQUAL  WEIGHT     .     .     .     .  22 

8.  SOLUTION  OF  NORMAL  EQUATIONS 24 

9.  THE  PROBABLE   ERROR      . 25 

10.  PROBABLE  ERRORS  FOR  INDIRECT  OBSERVATIONS  ....  29 

11.  PROBABLE  ERRORS  OF  COMPUTED  VALUES 31 

12.  CRITICAL  REMARKS  ....     .-'•• 33 

CHAPTER  II. 

PRECISE   PLANE  TRIANGULATIOM. 

13.  COORDINATES  AND  AZIMUTHS 36 

14.  MEASUREMENT  OF   ANGLES      :.     .     -.   ' 39 

15.  PROBABLE  ERRORS  AND  WEIGHTS  OF  ANGLES 43 

1 6.  THE  STATION  ADJUSTMENT      .     .'    .     .     .     .     .     *•    «,-     .  46 

17.  ERRORS  IN  A  TRIANGLE     .   v    .     •     .     .     .     ..    •     »     .  49 

18.  ADJUSTMENT  OF  A  TRIANGLE  .     .     .     .     .     .     .     .     ;     .  51 

19.  TRIANGLE  COMPUTATIONS    .  t  «     •     .    v    * 53 

20.  Two  CONNECTED  TRIANGLES 57 

3 


4  CONTENTS. 

PACK 

21.  DIRECT  OBSERVATIONS  WITH  ONE  CONDITION 59 

22.  INTERSECTIONS  ON  A  SECONDARY  STATION 61 

23.  THE  THREE-POINT  PROBLEM .64 

24.  GENERAL    CONSIDERATIONS 67 

CHAPTER   III. 

BASE   LINES. 

25.  PRINCIPLES  AND  METHODS 71 

26.  PROBABLE  ERROR  AND  UNCERTAINTY      .     .     .     »     .     .     .  72 

27.  BASES  AND  ANGLES .74 

28.  STANDARD  TAPES .<.'.'..'.     .     .  76 

29.  MEASUREMENT  WITH  A  TAPE .     .  79 

30.  BROKEN  BASES .     .'..»•'.     .     .  83 

31.  REDUCTION  TO  OCEAN    LEVEL 84 

CHAPTER   IV. 
LEVELING. 

32.  SPIRIT  LEVELING      .     .     .    ':     .     .     .    ..     /.'. .     .     .     .  87 

33.  DUPLICATE  LINES      .      .      .     ...     .     .     .     .     .     .     .  89 

34.  PROBABLE  ERRORS  AND  WEIGHTS 91 

35.  ADJUSTMENT  OF  A  LEVEL  NET .  94 

36.  GEODETIC    SPIRIT   LEVELING 96 

37.  REFRACTION  AND  CURVATURE \.     .     .  99 

38.  VERTICAL  ANGLES     .     .     .     ...     .     .     .     ...     .101 

39.  LEVELING  BY  VERTICAL  ANGLES 104 

CHAPTER  V. 
ASTRONOMICAL  WORK. 

40.  FUNDAMENTAL   NOTIONS       ......;-.-. 107 

41.  AZIMUTH  WITH   THE  SOLAR  TRANSIT no 

42.  AZIMUTH  BY  AN  ALTITUDE    OF  THE  SUN 114 

43.  AZIMUTH  BY  POLARIS  AT  ELONGATION 117 

44.  AZIMUTH  BY  POLARIS  AT  ANY  HOUR-ANGLE 121 

45.  LATITUDE  BY  THE   SUN 124 


CONTENTS.  5 

PAGE 

46.  LATITUDE  rv  A  STAR 127 

47.  TIME    .      .     .     .,     .     .   |f?,  .   ...     .     ,     »    %  _  .     •     .     .    129 

48.  LONGITUDE     .     .     .     .      .4, 132 

49.  PRECISE    DETERMINATIONS ".•   •'."     .134 

CHAPTER  VI. 
SPHERICAL  GEODESY. 

50.  EARLY  HISTORY  ....      .  ,,  .     ........  137 

51.  HISTORY  FROM   1300  TO  1750 .     .  140 

52.  MEASUREMENT  OF  MERIDIAN  ARCS 144 

53.  THE  EARTH  AS  A  SPHERE 147 

54.  LINES  ON   A  SPHERE •  .     .     .'    .     .     .  149 

55.  ANGLES   AND  TRIANGLES     .      .     .      .     .     *     .•     .     .     .     .  151 

56.  LATITUDES,    LONGITUDES,  AND  AZIMUTHS 153 

CHAPTER  VII. 

SPHEROIDAL   GEODESY." 

57.  PROPERTIES  OF  THE   ELLIPSE .  156 

58.  DISCUSSION  OF   MERIDIAN   ARCS  .,-...    ...     .     .     .     .  158 

59.  PLUMB-LINE   DEFLECTIONS.     ,     .     .........  164 

60.  DIMENSIONS  OF  THE   SPHEROID 168 

61.  LENGTHS  OF   MERIDIAN   AND   PARALLEL  ARCS       .     .      .     .170 

62.  NORMAL  SECTIONS  AND  GEODESIC  LINES 172 

63.  TRIANGLES  AND  AREAS.     .     .     .     .  •  .     .     .     ..-    .     .     .  175 

64.  LATITUDES,  LONGITUDES,   AND   AZIMUTHS    .     .     .     .*..-.  177 

CHAPTER  VIII. 

GEODETIC   COORDINATES   AND   PROJECTIONS. 

65.  THE   COORDINATE   SYSTEM.     .     .     ...     .     *     .     .     .  181 

66.  LMZ  COMPUTATIONS     .     .     .     .     ,     .     ^     .     ,     »     •     .  183 

67.  THE  INVERSE  LMZ  PROBLEM 186 

68.  MAP  PROJECTIONS     .      .      .     .     .     .     . 189 

69.  THE  POLYCONIC   PROJECTION *     .  192 

70.  RECTANGULAR   SPHERICAL    COORDINATES 194 


O  CONTENTS. 

CHAPTER  IX. 

GEODETIC   TRIANGULATION. 

PAG3 

71.  RECONNAISSANCE.     ". 198 

72.  STATIONS  AND  TOWERS 200 

73.  SIGNALS 204 

74.  HORIZONTAL  ANGLES    ...... 206 

75.  THE   STATION  ADJUSTMENT 210 

76.  TRIANGLE   COMPUTATIONS 212 

77.  THE  FIGURE   ADJUSTMENT 216 

78.  CONDITIONED  OBSERVATIONS 219 

79.  ADJUSTMENT  OF  A  POLYGON 222 

80.  ADJUSTMENT  OF  A  QUADRILATERAL .     .225 

81.  FINAL  CONSIDERATIONS 228 

CHAPTER  X. 
THE   FIGURE   OF   THE   EARTH. 

82.  THE  EARTH  AS  A  SPHEROID 232 

83.  THE  EARTH  AS  AN  ELLIPSOID 235 

84.  THE  EARTH   AS  AN   OVALOID 239 

85.  THE  EARTH  AS  A  GEOID 242 

86.  CONCLUSION 245 

CHAPTER  XI. 
TABLES. 

87.  EXPLANATION  OF  THE  TABLES 249 

I.     MEAN  CELESTIAL  REFRACTION 251 

II.     LENGTHS  OF  ARCS  OF  THE  MERIDIAN    .     .     .     .     .     .     .252 

III.  LENGTHS  OF  ARCS  OF  PARALLELS 253 

IV.  LOGARITHMS  FOR  GEODETIC  COMPUTATIONS 254 

V.     LOGARITHMS  FOR  THE  LMZ  PROBLEM      .     „     .     .     .     .255 

VI.     CONSTANTS  AND  THEIR  LOGARITHMS 256 

INDEX 257 


PRECISE  SURVEYING  AND  GEODESY. 


CHAPTER    I. 
THE    METHOD    OF    LEAST    SQUARES. 

1.  ERRORS  OF  OBSERVATIONS. 

The  Method  of  Least  Squares  furnishes  processes  of  com- 
putation by  which  the  most  probable  values  of  quantities  are 
found  from  the  results  of  measurements.  The  simplest  case 
is  that  of  a  quantity  which  is  directly  measured  several  times 
with  equal  precision ;  here  it  is  universally  agreed  that  the 
arithmetic  mean  of  the  several  values  is  the  most  probable 
value  of  the  quantity. 

When  a  quantity  is  measured  the  result  of  the  operation  is 
a  numerical  value  called  an  observation.  If  Z  be  the  true 
value  of  a  quantity  and  J/t  and  J/a  be  two  observations  upon 
it,  then  Z  —  Ml  and  Z  —  M^  are  the  errors  of  those  ob- 
servations. 

Constant  or  systematic  errors  are  those  which  result  from 
causes  well  understood  and  which  can  be  computed  or 
eliminated.  As  such  may  be  mentioned:  theoretical  errors, 
like  the  effects  of  refraction  upon  a  vertical  angle,  or  the 
effects  of  temperature  upon  a  steel  tape,  which  can  be  com- 
puted when  proper  data  are  known  and  hence  need  not  be 
classed  as  real  errors;  instrumental  errors,  like  the  effects  of 

7 


«     «« 

A  s:-?*-:*--*^ 

8  THE   METHOD   OF   LEAST   SQUARES.  I. 

an  imperfect  adjustment  of  an  instrument,  which  can  be 
removed  by  taking  proper  precautions  in  advance;  and  per- 
sonal errors  which  are  due  to  the  habits  of  the  observer,  who 
may,  for  example,  always  give  the  reading  of  a  scale  too 
great.  All  these  causes  are  to  be  carefully  investigated  and 
the  resultant  errors  removed  from  the  final  observations. 

Mistakes  are  errors  due  to  such  serious  mental  confusion 
that  the  observation  cannot  be  regarded  with  any  confidence, 
as  for  instance  writing  53  when  35  is  intended.  Observa- 
tions affected  with  mistakes  must  be  rejected,  although  when 
these  are  of  small  magnitude  it  is  sometimes  not  easy  to 
distinguish  them  from  errors. 

Accidental  errors  are  those  that  still  remain  after  all  con- 
stant errors  and  mistakes  have  been  carefully  investigated 
and  eliminated.  Such,  for  example,  are  the  errors  in  leveling 
arising  from  sudden  expansions  of  the  bubble  and  standards, 
or  from  the  effects  of  the  wind,  or  from  irregular  refraction. 
They  also  arise  from  the  imperfections  of  human  touch  and 
sight,  which  render  it  difficult  to  handle  instruments  delicately 
or  to  read  verniers  with  perfect  accuracy.  These  are  the 
errors  that  exist  in  the  final  observations  and  whose  discus- 
sion forms  the  subject  of  this  chapter. 

However  carefully  the  measurements  be  made,  the  final 
observations  do  not  agree;  all  of  these  observations  cannot 
be  correct,  since  the  quantity  has  only  one  value,  and  each 
of  them  can  be  regarded  only  as  an  approximation  to  the 
truth.  The  absolutely  true  value  of  the  quantity  in  question 
cannot  be  ascertained,  but  instead  of  it  one  must  be  deter- 
mined, derived  from  the  combination  of  the  observations, 
which  shall  be  the  "  most  probable  value,"  that  is  to  say,  the 
value  which  is  probably  nearest  to  the  true  value. 

The  difference  between  the  most  probable  value  of  a 
quantity  and  an  observation  is  called  the  *  residual  error  '  of 
that  observation.  Thus,  if  z  be  the  most  probable  value  of 


2.  LAW   OF   PROBABILITY   OF   ERROR.  9 

a  quantity  derived  from  the  observations  Ml  and  M»  and  vl 
and  vt  be  the  residual  errors,  then 

vl  =  z  -  Mi,      v,  =  z  -  Mt.  (i) 

When  the  measurements  are  numerous  and  precise  the  most 
probable  value  z  does  not  greatly  differ  from  the  true  value 
Z,  and  the  residuals  do  not  greatly  differ  from  the  true 
errors. 

Prob.  i.  Eight  measurements  of  a  line  give  the  values  186.4, 
186.3,  186.2,  186.3,  l86-3,  186.2,  185.9  an(*  186.4  inches,  and  its  most 
probable  length  is  their  arithmetic  mean.  Compute  the  eight 
residual  errors  ;  find  the  sum  of  the  positive  residuals  and  the  sum 
of  the  negative  residuals. 

2.  LAW  OF  PROBABILITY  OF  ERROR. 

The  probability  of  an  error  is  the  ratio  of  the  number  of 
errors  of  that  magnitude  to  the  total  number  of  errors.  If 
there  be  100  observations  of  an  angle  which  give  27  errors 
lying  between  i"  and  2"  the  probability  that  an  error  lies 
between  these  limits  is  0.27.  Probabilities  are  thus  measured 
by  numbers  lying  between  o  (impossibility)  and  I  (certainty). 

A  marksman  firing  at  a  target  with  the  intention  of  hitting 
the  center  may  be  compared  to  an  observer,  the  position  of 
a  shot  on  the  target  to  an  observation,  and  its  distance  from 
the  center  to  an  error.  If  the  marksman  be  skilled  and  all 
horizontal  errors,  like  the  effect  of  gravity,  be  eliminated  in 
the  sighting  of  the  rifle,  it  is  recognized  that  the  deviations 
of  the  shots,  or  errors,  are  quite  regular  and  symmetrical. 
First,  it  is  noticed  that  small  errors  are  more  frequent  than 
large  ones;  secondly,  that  errors  on  one  side  are  about  as 
frequent  as  on  the  other;  and  thirdly,  that  very  large  errors 
do  not  occur.  Moreover,  it  is  known  that  the  greater  the 
skill  of  the  marksman  the  nearer  are  his  shots  to  the  center  of 
the  target. 

As  an  illustration    a    record  of  one   thousand  shots  fired 


10 


THE    METHOD   OF   LEAST   SQUARES. 


I. 


from  a  battery  gun  at  a  target  six  hundred  feet  distant  may 
be  considered.  The  target  was  a  rectangle  fifty-two  feet  long 
by  eleven  feet  high,  and  the  point  of  aim  was  its  central  hori- 
zontal line.  All  the  shots  struck  the  target,  and  the  record 
of  the  number  in  the  eleven  horizontal  divisions,  each  one 
foot  in  width,  is  as  follows: 


In  top  division 
In  second  division 
In  third  division 
In  fourth  division 
In  fifth  division 
In  middle  division 
In  seventh  division 
In  eighth  division 
In  ninth  division 
In  tenth  division 
In  bottom  division 

Total 


1  shot 
4  shots 

10  shots 

89  shots 

190  shots 

212  shots 

204  shots 

193  shots 

79  shots 

16  shots 

2  shots 

i  ooo  shots 


The  figure  shows  by  means  of  ordinates  the  distribution  of 
these  shots;  A  being  the  top  division,  O  the  middle,  and  B 
the  bottom  division  It  will  be  observed  that  there  is  a 

slight  preponderance  of  shots 
below  the  middle,  and  there 
is  reason  to  believe  that  this 
is  due  to  a  constant  error  of 
gravitation  not  entirely  elim- 
inated in  the  sighting  of  the 
gun.  If  this  series  of  shots 

A  Q — ' — ' — ' — ' — £    were  to    be    repeated    again 

under  exactly  similar  condi- 
tions, it  might  be  fair  to  infer  that  0.212  would  be  the  prob- 
ability of  a  given  shot  striking  the  middle  division,  and  that 
o.ooi  would  be  the  probability  of  striking  the  top  division. 
Thus  the  probability  of  an  error  decreases  with  the  magnitude 
of  the  error, 


2. 


LAW   OF   PROBABILITY   OF   ERROR. 


II 


In  treatises  on  the  Method  of  Least  Squares  the  theory  of 
mathematical  probability  isf  applied  to  the  deduction  of  the 
relation  between  an  error  x,  and  its  probability  y.  The 
equation  deduced  is 

y  =  «-*-,  (2) 

where  c  and  h  are  constants  that  depend  upon  the  precision 
of  the  measurements  and  e  is  the  number  2.71828-  •.  This 
equation  expresses  the  law  of  probability  of  accidental  errors 
of  observations.  It  shows  that  y  has  its  greatest  value  when 


x  is  zero,  that  y  becomes  very  small  when  x  is  very  large, 
and  that  the  same  value  of  y  is  given  by  equal  positive 
and  negative  values  of  x.  The  figure  shows  the  curve 
expressed  by  the  equation,  x  and  y  being  parallel  to  the  axes 
OX  and  OY,  and  OM  being  any  error  whose  probability  y  is 
given  by  the  ordinate  MN. 

This  law  is  deduced  under  the  supposition  of  a  very  large 
number  of  errors,  and  hence  in  a  particular  case  close  agree- 
ment is  not  to  be  expected.  For  any  series  of  errors  the 
values  of  c  and  //*  can  be  computed  and  the  theoretic  number 
of  errors  can  then  be  compared  with  those  actually  observed. 
For  example,  in  the  above  case  of  the  shots  on  the  target, 
the  value  of  c  can  be  found  to  be  0.234  and  that  of  h*  to  be 
0.173,  and  the  following  comparison  shows  the  agreement  of 
practice  with  theory: 


For  division  No.  i  2  3  45  6  7  8  91011 
Actual  shots  i  4  10  89  190  212  2^4  193  79  16  2 
Theoretic  shots  3  15  50  118  197  234  197  118  50  15  3 

The  dotted  curve  on  the  graphic  representation  shows  the 
theoretic  distribution  of  the  shots.  In  general  it  may  be  said 


12  THE   METHOD   OF  LEAST   SQUARES.  I. 

that  the  results -of  observation  are  in  good  agreement  with 
the  theoretic  law,  and  that  this  agreement  is  closer  the  greater 
the  number  of  errors  considered. 

Prob.  2.  Given  the  equation  y  =  o.234<r~°-173^a  to  compute  the 
values  of  y  when  the  values  of  x  are  i,  2,  3,  4  and  5. 

3.    THE  PRINCIPLE  OF  LEAST  SQUARES. 

The  Method  of  Least  Squares  sets  forth  the  processes  by 
which  the  most  probable  values  of  observed  quantities  are 
derived  from  the  observations.  The  foundation  of  the 
method  is  the  following  principle: 

In  observations   of  equal    precision    the  most   probable 
values  of  observed  quantities  are  those  that  render  the 
sum  of  the  squares  of  the  residual  errors  a  minimum. 
This    principle  was    first  enunciated    by  Legendre    in    1805, 
and  has  since  been  universally  accepted  and  used  as  the  basis 
of  the  science  of  the  adjustment  of  observations.      Starting 
with  the  law  of  probability  of  error  enunciated   <'n   the  last 
Article  it  may  be  proved  in  the  following  manner: 

Let  n  observations  result  from  measurements  of  equal  pre- 
cision, so  that  c  and  //  in  (2)  are  the  same  in  each  case. 
Then  the  probability  of  the  error  xi  in  the  first  observation 
is  ce-^^y  that  of  the  error  ;ra  is^-*9jr»a,  and  that  of  the 
error  xn  is  ce~h*x**.  If  the  observations  be  independent  of 
each  other,  the  probability  of  the  simultaneous  occurrence  of 
the'  errors  x^  x^  .  .  .  xn  is  the  product  o/  *heir  respective 
probabilities,  or  that  probability  is 

p  _  cne-  h\*s  +  .*,«  +  ...+  x*\t 

Now  the  true  values  of  the  errors  x^  x^  .  .  .  xu  and  those  of 
the  quantities  observed  cannot  be  found,  but  the  best  that 
can  be  done  is  to  find  their  most  probable  values,  namely  the 
values  that  give  the  greatest  probability  P.  The  greatest 
value  of  P  occurs  when  x*  -\-  x*  +  •  •  •  +  *«*  has  its  least 


3.  THE   PRINCIPLE    OF   LEAST   SQUARES.  13 

value,  that  is,  the  most  probable  values  are  those  that  render 
the  sum  of  the  squares  of  the  errors  a  minimum.  Let  vl  , 
v9 ,  .  .  .  vn  represent  the  most  probable  values  of  the  errors 
xlt  *a ,  .  .  .  xn ,  then- 

v*  -f-  v*  +  •  •  •  +  vn  =  a  minimum  (3) 

is  an  algebraic  statement  of  the  fundamental  principle  of  least 
squares. 

An  application  of  this  principle  to  the  common  case  of 
direct  observations  on  a  single  quantity  will  now  be  given. 
Let  z  be  the  most  probable  value  of  the  quantity  whose  n 
observed  values  are  M^ ,  M9 ,  .  .  .  MH ,  all  being  of  equal  pre- 
cision. Then  the  residual  errors  are 

vl  =  z  —  Mlt     z/,  =  *  —  M,,  .  .  .  vn  =  z  —  Mn\ 
and  from  the  fundamental  principle  (3), 
(z  —  J/,)J  +  (z  —  M^f  -f-  .  .  .  +  (z  —  Mn)*  =  a  minimum. 

To  find  the  value  of  z  which  renders  this  expression  a  mini- 
mum it  is  to  be  differentiated  and  the  derivative  placed  equal 
to  zero,  giving 

2(2  —  Mt)  -f-  2(2  —  MI)  .  .  .  -\-\2(z  —  Mn)  =  o, 
from  which  the  value  of  z  is  found  to  be 

_._Jfl  +  Jlf.+  ...+Mn     t 
n 

that  is,  the  most  probable  value  of  the  observed  quantity  is 
the  arithmetic  mean  of  the  observations. 

It  has  been  universally  recognized  from  the  earliest  times 
that  the  arithmetic  mean  gives  the  most  probable  value  of  a 
quantity  which  has  been  measured  several  times  with  equal 
care.  Indeed  some  authors  have  regarded  this  as  an  axiom 
and  used  it  to  deduce  the  law  of  probability  of  error  stated 
in  equation  (2).  It  should  be  noted  that  the  method  of  the 
arithmetic  mean  only  applies  to  equally  good  observations  on 
a  single  quantity,  and  that  it  cannot  be  used  when  observa- 


14  THE   METHOD    OF   LEAST   SQUARES.  I. 

tions  are  made  on  several  related  quantities.  For  instance, 
let  an  angle  be  measured  and  found  to  be  6o£  degrees,  and 
again  be  measured  in  two  parts,  one  being  found  to  be  40 
and  the  other  2O  degrees.  The  proper  adjusted  value  of  the 
angle  is  not,  as  might  at  first  be  supposed,  the  mean  of  60^ 
and  60,  which  is  6oJ  degrees,  but,  as  will  be  seen  later,  it  is 
6oJ  degrees,  a  result  which  requires  each  observation  to  be 
corrected  the  same  amount. 

Prob.  3.  Four  measurements  of  a  base  line  give  the  observations 
1472.34  feet,  1471.99  feet,  1472.25  feet,  and  1472.14  feet.  Compute 
the  sum  of  the  residual  errors,  arid  the  sum  of  .the  squares  of  the 
residuals. 

4.  WEIGHTED  OBSERVATIONS. 

Observations  have  equal  precision  when  all  the  measure- 
ments are  made  with  the  same  care,  or  when  no  reason  can 
be  assigned  to  suppose  that  one  is  more  reliable  than  another: 
they  are  then  said  to  have  equal  "  weight."  Weights  are 
numbers  expressing  the  relative  practical  worth  of  observa- 
tions, so  that  an  ,  observation  of  weight/  is  worth/  times  as 
much  as  an  observation  of  weight  unity.  Thus  if  a  line  be 
measured  five  times  with  the  same  care,  three  measurements 
giving  950.6  feet  and  two  giving  950.4  feet,  then  the  numbers 
3  and  2  are  the  weights  of  the  observations  950.6  and  950.4. 
Thus  "  950.6  with  a  weight  of  3  "  expresses  the  same  as  the 
number  950.6  written  down  three  times  and  regarded  each 
time  as  having  a  weight  of  unity;  or  "950.6  with  a  weight 
of  3  "  might  mean  that  950.6  is  the  arithmetic  mean  of  three 
observations  of  weight  unity. 

Let  Ml ,  M^ ,  .  .  .  Mn  be  n  observations  made  upon  quan- 
tities whose  most  probable  values  are  to  be  found.  Let  the 
residual  errors  be  v^ ,  v^ ,  .  .  .  vn  ,  and  by  the  principle  of  least 
squares  the  values  to  be  found  for  the  quantities  must  be 
such  that 

v?  -f~  v*  +  •  •  •  H~  VH  =  a  minimum. 


4.  WEIGHTED    OBSERVATIONS.  15 

Now  suppose  that  there  are/!  observations  having  the  value 
Ml ,  or  that  Ml  has  the  weight  /, ;  also  that  M^  and  Ms  have 
the  weights /a  and/,.  Then  there  will  be'/,  residuals  having 
the  value  v^  ,  /2  having  the  value  v^ ,  and  /,  having  the 
value  v%.  Thus  the  condition  becomes 

P^i  +  PJ>?  +  •  •  •  +  p»vH*  =  a  minimum.         (4) 
Hence  a  more  general  statement    of  the    principle  of  least 
squajj£§  is: 

In  observations  of  unequal  precision  the  most  probable 
values  of  the  observed  quantities  are  those  which 
render  the  sum  of  the  weighted  squares  of  the  residual 
errors  a  minimum. 
Here  it  is  seen  that  the  term  "  weighted  square  of  a  re- 
sidual "  means  the  product  of  the  square  of  the  residual 
by  its  weight. 

An  application  of  this  principle  to  the  case  of  weighted 
observations  on  a  single  quantity  will  now  be  made.  Let  z 
be  the  most  probable  value  of  the  quantity  whose  observed 
values  are  M1  ,  J/, ,  .  .  .  Mn  having  the  weights/,  ,/,,..  ./„. 
Then  the  residuals  are  z  —  Ml ,  z  —  M9 ,  .  .  .  z  —  Mn ,  and 
from  the  general  principle  of  least  squares  given  by  (3), 
p,(z  -  M^+pte  -  M$  +  .  .  .  +/„(>  -  MH)*  =  a  minimum. 
The  first  derivative  of  this,  placed  equal  to  zero,  gives 

/,(*  -  M,)  +ft,(g  -M,)+  .  .  .  +fm(s  -  M,)  =  o, 
from  which  the  most  probable  value  of  z  is 
j».A 

The  value  of  z  thus  found  is  sometimes  called  the  weighted 
arithmetic  mean,  and  the  method  of  computing  it  is  fre- 
quently expressed  by  the  rule:  Multiply  each  observation  by 
its  weight  and  divide  the  sum  of  the  products  by  the  sum  of 
the  weights. 

Prob.  4.  Prove  the  principle  (4)  directly  from  the  law  of  prob- 


l6  THE    METHOD    OF   LEAST   SQUARES.  I. 

ability  of  error  given  by  (2),  assuming  that  J?  represents  the  weight 
of  the  observation  whose  error  is  x. 

5.  OBSERVATION  EQUATIONS. 

When  observations  are  taken  of  several  related  quantities, 
the  measurements  are  usually  made  upon  functions  of  those 

quantities.  Thus  the  sum  and  differ- 
ence of  two  quantities  might  be  ob- 
served instead  of  the  quantities  them- 
selves.  Such  measurements  produce 
"indirect  observations"  which  are 
generally  represented  by  equations 
called  "observation  equations."  To 
illustrate  how  they  arise,  let  the  following  practical  case  be 
considered.  Let  O  represent  a  bench-mark,  and  X,  Y,  Z, 
three  points  whose  elevations  above  O  are  to  be  determined. 
Let  five  lines  of  levels  be  run,  giving  the  results 

Observation  i.  X  above  O  =  10.35  ^eet' 

Observation  2.  Y  above  X  =     7.25  feet. 

Observation  3.  Y  above  O  =  17.63  feet. 

Observation  4.  Y  above  Z  =     9.10  feet. 

Observation  5.  Z  below  X  =     1.94  feet. 

Here  it  will  be  at  once  perceived  that  the  measurements  are 
discordant;  if  observations  I  and  2  are  taken  as  correct,  the 
elevation  of  X  is  10.35  feet,  and  that  of  Y  is  17.60  feet;  if  2 
and  3  are  correct,  then  X  is  10.38  feet  and  Y  is  17.63  feet; 
and  it  will  be  found  impossible  to  deduce  values  that  will 
exactly  satisfy  all  the  observations.  Let  the  elevations  of 
the  points  X,  Y,  and  Z  above  O  be  denoted  by  ;r,  y,  and  s, 
then  the  observations  furnish  the  following  equations: 

1.  *==  10.35, 

2.  y-x=     7.25, 

3.  y  =  17.63, 

4.  y-  z  =    9.10, 

5.  x  —  z=     1.94. 


5.  OBSERVATION   EQUATIONS.  1 7 

The  number  of  these  equations  is  five,  the  number  of  the 
unknown  quantities  is  th,re~e,  and  hence  an  exact  solution 
cannot  be  made.  The  best  j;hat  can  be  done  is  to  find  values 
for  x,  y,  and  z  which  are  the  most  probable,  and  these  will 
be  found  in  the  next  Article  by  the  help  of  the  principle  of 
least  squares. 

Observations  are  called   "  direct  "   when   made  upon   the  1 
quantity  whose  value  is  sought,  and  "  indirect  "  when  made  \ 
upon   functions  of  the  quantities  whose  values  are  required.  J 
Thus  in  the  above  example  the  first  and  third  observations 
are   direct,   and   the   others   are    indirect,    being  made   upon 
differences  of  elevation  instead  of  upon  the  elevations  them- 
selves.     Indirect  observations  are  of  frequent  occurrence  in 
the  operations  of  precise  surveying. 

Quantities  are  said  to  be  "  independent"  when  each  can 
vary  without  affecting  the  value  of  the  others;  thus  in  the 
above  example  the  elevation  of  any  one  station  above  the 
bench-mark  O  is  entirely  independent  of  the  elevations  of  the 
others,  or  in  other  words  there  is  no  necessary  relation 
between  the  values  of  x^  y,  and  z. 

Quantities  are  said  to  be  "  conditioned  "  when  they  are  so  1 
related  that  a  change  in  one  necessarily  affects  the  values  of  \ 
the  others;  thus  if  the  three  angles  of  a  plane  triangle  be 
called  x,  y,  and  zt  it  is  necessary  that  x  -\-  y  -f-  s  =  180°  and 
the  values  to  be  found  for  the  angles  must  satisfy  this  condi- 
tion. In  stating  observation  equations  it  will  often  be  found 
best  to  select  the  quantities  to  be  determined  in  such  a  way 
that  they  shall  be  independent;  thus  if  the  three  angles  of 
a  triangle  are  observed  to  be  62°  20'  43",  36°  14'  06",  and 
81°  25'  08",  let  x  and  y  denote  the  most  probable  values  of 
the  first  and  second  angles,  then  the  observation  equations 
are 

x  =  62°  20'  43", 
y  —  36    14  06, 
180°  —  x  —  y  —  81    25  08, 


1  8  THE  .^METHOD    OF   LEAST   SQUARES.  I. 

the  last  of  which  may  be  written 

98°34'  52", 


and  here  x  and  y  are  independent  quantities.  Thus  by 
properly  limiting  the  number  of  unknown  quantities  these 
can  always  be  rendered  independent  of  each  other. 

As  a  second  example  of  the  statement  of  observation 
equations  take  the  following  values  of  the  angles  measured 
at  North  Base,  Keweenaw  Point,  on  the  United  States  Lake 
Survey  : 

CNM=.    55°  57'  58".68, 

MNQ=    48    49    13   .64, 

CNQ  =  104   47    12   .66, 
QNS=    54    38    15   .53, 

MNS  =  103    27   28   .99. 

The  object  of  these  observations  is  to  find  the  values  of  the 
four  angles  around  the  point  N\  but  if  x^  y,  z>  and  w  repre- 
sent these  angles,  then  x-\-y-\-z-\-w=  360°  and  the 
quantities  are  conditioned.  To  make  the  quantities  inde- 
pendent only  three  unknowns  should  be 
taken;  thus  let  CNM  —  x,  MNQ  —  y,  and 
QNS  =  2,  then  the  observation  equations  are 

*=    55°  57'  58".68, 
y  =    48    49    13   .64, 
=  104   47    12   .66, 

~    54    38    15   -53, 
=  103    27    28  .99, 

and  in  the  next  Article  it  will  be  shown  how  the  most  prob- 
able values  of  x,  y,  and  z  are  to  be  found. 

Thus,  in  general,  observations  upon  several  quantities  lead 
to  observation  equations  whose  number  is  greater  than  that 
of  the  unknown  quantities,  and  no  system  of  values  can  be 
found  that  will  exactly  satisfy  the  observation  equations. 
They  may,  however,  be  approximately  satisfied  by  many 


6.  INDIRECT   OBSERVATIONS   OF  EQUAL  WEIGHT.  19 

systems  of  values;    and  the    problem  is  to  determine  that 
system  which  is  the  most  probable  and  hence  the  best. 

ProK  5.  State  observation  equations  for  the  above  example,  tak- 
ing SNQ  =  j,  SNM-=  t,  Stfb  =  u. 


6.  INDIRECT  OBSERVATIONS  OF  EQUAL  WEIGHT. 
When  observation  equations  have  been  written  so  that  the 
unknown  quantities  have  no  necessary  relation  to  each  other, 
the  case  is  called  that  of  indirect  independent  observations. 
Let  Ml  ,  My  ,  .  .  .  Mn  be  n  observations  of  equal  weight  made 
upon  functions  of  the  unknown  quantities  x,  y,  z,  etc.  Let 
the  observations  give  the  following  observation  equations: 


anx  -\-  bny  +  cnz  +  .  .  .  =  Mn  , 

in  which  at  ,  a^  ,  .  .  .  an  ,  bl  ,  b^  ,  .  .  .  bn  ,  etc.,  are  known  co- 
efficients of  the  unknown  quantities.  The  most  probable 
values  of  x,  y,  z^  etc.,  when  found  and  inserted  in  the 
equations  will  not  exactly  satisfy  them,  but  leave  small  resid- 
ual errors,  vl  ,  z/a  ,  .  .  .  vn  ;  thus  strictly 

a^x  -f-  b^y  -\-  c^z  -\-  .  .  .  —  M^  =  i\  , 


anx  +  bn  y  +  cnz  +  •  •  •  —  Mn  —  vn , 

and,  by  the  principle  of  least  squares  given  by  (2)  in  Art.  2, 
the  sum  of  the  squares  of  these  residuals  must  be  a  minimum 
in  order  to  give  the  most  probable  values  of  xy  y,  and  z. 

In  order  to  find  the  condition  for  the  most  probable  value 
of  x  let  the  terms  independent  of  x  in  the  equations  be 
denoted  by  N^ ,  Nt ,  .  .  .  Nn ;  then  they  may  be  written 


anx  +  Nn  =  v 


2O  THE   METHOD   OF   LEAST   SQUARES.  I. 

Squaring  both  terms  of  these  equations,  and  adding,  gives 

J  +  .  .  .  +  (anx  + 


and  this  is  to  be  made  a  minimum  to  give  the  most  probable 
value  of  x.  Differentiating  it  with  respect  to  x,  and  placing 
the  first  derivative  equal  to  zero,  there  results 


«»,(«,*  +  Nt)  +  a,(a^  +  /V,)  +  .  .  .  +  a,(anX  +  TV.)  =  o,     (6) 

and  this  is  the  condition  for  the  most  probable  value  of  x. 
In  like  manner  a  similar  condition  may  be  stated  for  each  of 
the  other  unknown  quantities.  The  conditions  thus  stated 
are  called  "  normal  equations,  "  and  their  solution  will  furnish 
the  most  probable  values  of  the  required  quantities. 

The  following  is  hence  the  rule  for  the  adjustment  of 
observations  of  equal  weight  involving  several  independent 
quantities: 

For  each  of  the  unknown  quantities  form  a  normal  equa- 
tion by  multiplying  each  observation  equation  by  the 
coefficient  of  that  unknown  quantity  in  that  equation 
and  adding  the  results.  Then  the  solution  of  these 
normal  equations  will  furnish  the  most  probable 
values  of  the  unknown  quantities. 

In  forming  the  normal  equations  it  should  be  particularly 
noticed  that  the  signs  of  coefficients  are  to  be  observed  in 
performing  the  multiplications,  and  also  that  when  the 
unknown  quantity  under  consideration  does  not  occur  in  an 
observation  equation  its  coefficient  is  o. 

As  an  example  the  five  observation  equations  at  the  begin- 
f  the  last  Article  will  be  taken.      They  may  be  written 

1.  x  =  10.35, 

2.  —  X+y  =      7.25, 

3--  y       =  17.63, 

4.  y  —  z  —     9.10, 

5.  x          —  *  =     1.94. 


6.  INDIRECT   OBSERVATIONS   OF    EQUAL   WEIGHT.  21 

Now  to  form  the  normal  equation  for  x>  the  first  equation  is 
to  be  multiplied  by  I,  the  second  by  —  I,  and  the  fifth  by 
I  ;  and  adding  these,  ^ 

~3*  -  y  -  *  =  5-04. 

In  like  manner  to  find  the  normal  equation  for  7,  the  second 
equation  is  multiplied  by  I,  the  third  by  i,  and  the  fourth 
by  i,  whence 

-   X  +   &   -  2  =    33.98. 

Lastly,  to  find  the  normal  equation  for  #,  the  fourth  equation 

is  multiplied  by  —  i  and  the  fifth  by  —  I,  and  adding,  ^ 

—  x  —  y  -\-  2z  —  —  II .04. 

These  three  normal  equations  contain  three  unknown  quan- 
tities, and  their  solution  gives 

x  =  10.372,         y  =r  17.61,          z  =  8.47  feet, 
which  are  the  most  probable  values  of  the  three  elevations. 

As  a  second  example  the  three  observation  equations  near 
the  middle  of  the  last  Article  are 

x  =     62°  20'  43", 

y  =     36     14    06, 
x  +    y  =     98     34     52. 
Applying  the  rule,  the  two  normal  equations  are 

2x  +    y  =  1 60     55     35, 
x  +  2y  =   134    48     58, 

and  the  solution  of  these  gives 

x  =.  62°  20'  44",     y  =  36°  14'  07", 

whence  the  third  angle  of  the  triangle  is  180  degrees  minus  the 
sum  of  these,  or  81°  25'  09".  By  comparing  these  with  the 
observed  values  it  will  be  seen  that  each  observation  is  cor- 
rected by  the  same  amount;  this  is  because  the  observations 
are  of  equal  weight  and  each  angle  is  similarly  related  to  the 
other  two. 


22  THE   METHOD    OF   LEAST    SQUARES.  I. 

Prob.  6.  Form  and  solve  the  normal  equations  for  the  observa- 
tion equations  of  Prob.  5. 

7.  INDIRECT  OBSERVATIONS  OF  UNEQUAL  WEIGHT. 

The  two  preceding  Articles  give  the  method  of  adjusting 
indirect  observations  of  equal  weight  upon  several  independ- 
ent quantities;  now  is  to  be  investigated  the  case  of  indirect 
observations  of  different  weights  upon  such  quantities.  Let 
/,  ,  /a  ,  .  .  .  pn  be  the  weights  of  the  n  observations  Ml  , 
M  ,  ...  Mn  ,  so  that  the  observation  equations  are 

a^x  +  b^y  +  c^z  -f-  .  .  .  =  Ml  ,      with  weight/,  , 
+  b,y  +  cjs  +  .  .  .  =  M,  ,      with  weight  /3  , 


anx  -f-  bny  -\-  cnz  -f-  .  .  .  =  Mn,     with  weight  pn. 

Now  if  the  first  equation  were  written/,  times,  the  second  /a 
times,  etc.,  all  the  equations  would  have  the  same  weight 
and  the  rule  of  the  last  Article  would  apply.  That  is,  if  each 
of  the  above  equations  be  multiplied  by  the  coefficient  of  x  in 
that  equation,  and  also  by  its  weight,  the  sum  will  be  the 
condition  for  the  most  probable  value  of  x\  and  in  like 
manner  is  found  the  condition  for  the  most  probable  value  of 
each  of  the  other  unknowns.  These  conditions  are  the 
normal  equations. 

The  following  is  hence  the  rule  for  the  adjustment  of 
observations  of  unequal  weight  upon  several  independent 
quantities: 

For  each  of  the  unknown  quantities  form  a  normal  equa- 
tion by  multiplying  each  observation  equation  by  the 
coefficient  of  that  unknown  quantity  in  that  equation, 
and  also  by  its  weight,  and  adding  the  results.      The 
solution  of    these   normal    equations  will  furnish    the 
most  probable  values  of  the  unknown  quantities. 
In  applying  this  rule  the  same  precautions  are  to  be  observed 
regarding  signs  of  the  coefficients  as  before  stated. 


7,  INDIRECT   OBSERVATIONS   OF   UNEQUAL   WEIGHT.          23 

An  algebraic  expression  of  the   normal    equations  can  be 
made  by  introducing  the  following  abbreviations: 

[X]  ==.  A*,"         '-k,  A".'  +  ,     •    •    + 

\_pab~]      =  //z  A     +  A^A     -[-  .  .  .  + 
[paM]  =  AM*i 


and  then  the  normal  equations  can  be  written 

\_pcf\x  +  \_pab~\y-\-  \_pac\s  +  etc.  —  \_paM~\, 


*  +  etc.  -  , 

+  ]Jbc\y  +  [X]«  +  etc.  -  ' 


Here  it  will  be  seen  that  the  coefficients  of  the  unknown 
quantities  in  the  first  vertical  column  are  the  same  as  those 
in  the  first  horizontal  line,  those  in  the  second  column  the 
same  as  those  in  the  second  line,  and  so  on.  This  is  a  char- 
acteristic of  normal  equations  and  serves  as  a  check  when  they 
are  deduced  by  direct  application  of  the  rule.  If  the 
observations  are  of  equal  weight,  p  is  to  be  made  unity 
throughout,  and  the  method  reduces  to  that  of  the  last 
Article. 

As  a  numerical  illustration  let  five   observations  produce 
the  five  observation  equations 

\.-\-x  =        o,  with  weight     3, 

2.  ~\~  y  —        °>  with  weight  19,       • 

3.  -f-  2  =        o,  with  weight  13, 

4.  +  x  +  y  —  +  0.34,      with  weight  17, 

5.  -f-  y  -f-  z  =  —  o.  18,      with  weight    6. 

From  these  the  normal  equations,  formed  either  by  the  rule 
or  by  help  of  the  algorithm,  are 

+  i?y  =  +  5.78, 

+  427  +    6s  =  +  4.70, 
—  —  i.  08, 


24  THE    METHOD    OF   LEAST   SQUARES.  I. 

whose  solution  furnishes  the  results 

x  =  +  0.285,     y  =  +  0.005,      *  =  —  0.059, 
which  are  the  most  probable  values  of  the  required  quantities. 

Prob.  7.  In  a  plane  triangle  six  observations  give  A  —  42°  17'  35", 
three  observations  give  B  =  56°  40'  09",  and  two  observations 
give  C  =  81°  02'  10".  Compute  the  adjusted  values  of  the  angles. 

8.   SOLUTION  OF  NORMAL  EQUATIONS. 

The  normal  equations  which  arise  in  the  adjustment  of 
observations  may  be  solved  by  any  algebraic  process.  It  is 
desirable,  however,  to  use  methods  which  will  furnish  the 
value  of  each  unknown  quantity  independently  of  the  others, 
as  the  liability  to  error  is  thus  lessened.  When  there  are  but 
two  normal  equations  let  them  be  expressed  in  the  form 

A,x  +  B,y  =  A  , 
A^  +  Bj  =  £>., 
then  the  solution  by  any  method  gives  the  formulas 

AA  -  AA  A  A  -  A  A 

~  B,A,  -  B,AC  ~  A  &  -  A,B^  ' 

which  can  easily  be  kept  in  mind  by  noting  the  order  of  the 
letters  and  subscripts.      It  may  be  observed  also  that  the  two 
denominators  are  equal  numerically  but  of  contrary  sign. 
For  three  normal  equations  let  them  be  written  in  the  form 
A,x  +  B,y  +  O  =  A  , 


and  the  solution  leads  to  the  formulas 

1 


~ 


-  (A.C.-AtC^+^C.-A.C^+^Ct-A^)*. 


Q.  THE    PROBABLE   ERROR.  25 

in  which  the  three  denominators  have  the  same  value.  After 
a  little  practice  it  will  be  easy  to  use  these  formulas  with 
great  rapidity  in  the  solutiori  of  normal  equations. 

When  the  number  of  normal  equations  is  greater  than 
three,  general  formulas  for  solution  are  too  lengthy  to  be 
written,  and  the  systematic  method  of  substitution  devised 
by  Gauss  is  generally  employed.  This  is  explained  and 
exemplified  in  text-books  on  the  Method  of  Least  Squares, 
but  lack  of  space  forbids  its  presentation  here. 

Prob.  8.  Solve  the  normal  equations 

3*  —  y  +  2z  =  5,      —x  +  4y  +  z=  6,     2* -f  y  +  53  =  3, 
and   check    the   solution    by  showing  that   the  values  of  the   roots 
satisfy  the  equations. 

9.   THE  PROBABLE  ERROR. 

The  Method  of  Least  Squares  comprises  two  tolerably  dis- 
tinct divisions.  The  first  is  the  adjustment  of  observations, 
or  the  determination  of  the  most  probable  values  of  observed 
quantities.  The  second  is  the  investigation  of  the  precision 
of  observations  and  of  the  adjusted  results.  The  first  is  done 
by  the  application  of  the  principle  of  least  squares  given  in 
Art.  3 ;  the  second  is  done  by  the  determination  of  the 
probable  error,  the  rules  for  which  will  now  be  presented. 

The  following  may  be  stated  as  a  definition  of  the  term 
"  probable  error  "  : 

In  any  large  series  of  errors  the  probable  error  is  an 
error  of  such  a  value  that  the  number  of  errors  less 
than  it  is  the  same  as  the  number  greater  than  it. 
The  probable  error  is  hence  an  error  which  is  as  likely  as 
not  to  be  exceeded.  In  the  figure  if  the  ordinate  MN  be 
drawn  so  as  to  divide  the  area  on  each  side  of  O  Y  into  two 
equal  parts,  then  OM  is  the  probable  error.  Here  the  total 
area  between  the  curve  and  the  Jf-axis  is  unity  (certainty), 
and  the  area  MN YNM  is  0.5  ;  thus  the  probability  that  an 


THE    METHOD    OF   LEAST   SQUARES. 


I. 


error  is  greater  than  OM  is  0.5,  and  that  it  is  less  than  OM 
is  also  0.5. 

To  render  more  definite   the  conception  of  probable  error 
let  two  sets  of  observations  made  upon  the  length  of  a  line 


X  MOM  X 

be  considered.  The  first  set,  made  with  a  chain,  gives  634.7 
feet  with  a  probable  error  of  0.3  feet.  The  second  set,  made 
with  a  tape,  gives  634.64  with  a  probable  error  of  0.06  feet; 
thus, 

/,  =  634.7  ±  0.3     and     /„  =  634.64  ±  0.06; 

and  it  is  an  even  chance  that  634.7  is  within  0.3  of  the  truth, 
and  also  an  even  chance  that  634.64  is  within  0.06  of  the 
truth.  The  probable  error  thus  gives  an  absolute  idea  of  the 
accuracy  of  the  results;  it  also  serves  as  a  means  of  compar- 
ing the  precision  of  different  observations,  for  in  the  above 
case  the  precision  of  the  second  result  is  to  be  taken  as  much 
cnrgflter  than  that  of  the  first. 

It  is  a  principle  of  the  Method  of  Least  Squares  that 
weights  of  observations  are  reciprocally  proportional  to  the 
squares  of  their  probable  errors.  Thus,  for  the  above  numeri- 
cal example, 

'™  I/25' 


Hence  the  second  observation  has  a  value  about  25  times 
that  of  the  first  when  it  is  to  be  used  in  combination  with 
other  measurements.  Weights  and  probable  errors  are  con- 
stantly used  in  the  discussion  of  observations.  Weights  are 
usually  determined  from  the  number  of  measurements  or  from 
knowledge  of  the  manner  in  which  they  are  made,  but  prob- 
able errors  are  computed  from  the  observations  themselves. 


9.  THE   PROBABLE    ERROR.         ,  2/ 

For  the  case  of  direct  observations  on  the  same  quantity, 
all  being  of  equal  precision^  the  arithmetic  mean  is  the  most 
probable  value  (Art.  3).  Subtracting  each  observation  from 
the  mean  gives  the  residuals  vl  ,  v^ ,  .  ,  .  vn ,  and  the  sum  of 
the  squares  of  these  is  represented  by  JSV2.  Then 


=  0.6745Y  —  -  (9) 

is  the  probable  error  for  a  single  observation,  and,  since  n  is 
the  weight  of  the  arithmetic  mean, 


is  the  probable  error  of  the  arithmetic  mean.  For  example, 
let  six  observations  of  an  angle  be  taken  with  equal  care  and 
let  these  be  arranged  as  below  in  the  column  headed  M. 
The  sum  of  these  values  divided  by  6  gives  48°  06'  14"  './  as 


M 

IS                          *U 

48°o6/  12' 

'.5          +  2".2             4.84 

15 

.0 

—  o 

.3         0.09 

r,  ==  2-58 

20 

•3 

-5 

.6       31.36 

08 

•9 

+  5 

.8      33.64 

z  =  48°  06' 

'5 

.1 

—  o 

.4        0.16 

16 

•4 

—  i 

.7         2.89 

r  =  i".os 


14".  7  ±  i 


z  =  48°  06'  14". 7  o".o       73.98  =J?f? 

the  most  probable  value  of  the  angle,  the  second  column 
gives  the  residuals,  and  the  third  their  squares.  Then  by 
the  use  of  the  formula  the  probable  error  of  a  single  observa- 
tion is  found  to  be  2". 58  and  that  of  the  arithmetic  mean,  to 
be  I/;.O5.  Thus  if  another  observation  were  to  be  taken  it 
is  as  likely  as  not  that  it  will  deviate  2^.58  from  the  truth. 

For  the  case  of  n  observations  of  different  weights  on  one 
quantity  the  weighted  mean  is  the  most  probable  value 
(Art.  9).  Subtracting  each  observation  from  this  gives  the 
residuals,  and  the  square  of  each  of  these  is  to  be  multiplied 


28  THE    METHOD   OF   LEAST   SQUARES.  1. 

by  its  weight  to  give  the  sum  of  the  weighted  squares,  which 
may  be  represented  by  2£pv*.     Then 


r,  =  0.6745^^  (9)" 

is  the  probable  error  of  an  observation  of  the  weight  unity. 
and  if  2f  represent  the  sum  of  the  weights, 

'    -      •  •    r=7F        " 

is  the  probable  error  of  the  weighted  mean.  As  an  example, 
let  the  observations  in  the  first  column  of  the  following  table 
be  the  results  of  the  repetition  of  an  angle  at  different  times, 
i8".26  arising  from  five  repetitions,  i6".3O  from  four,  and 
so  on,  the  weights  of  the  observations  being  taken  the  same 
as  the  number  of  repetitions.  Then  the  general  mean  z  has 

M  p 

32°  of  i8".26  5 

16    .30  .  4 

21      .06  I 

'7    -95  4 

16   .20  3 

20    .85  4 


V 

Z/* 

pv> 

— 

o".io 

O.OIO 

0.05 

+ 

i    .86 

3.460 

13-84 

— 

2      .90 

8.410 

8.41 

+ 

0     .21 

0.044 

0.18 

+ 

I      .96 

3.842 

"•53 

*•*. 

2      .69 

7.236 

28.94 

z  —  32°  07'  i8".i6  2i=2f 

the  weight  21,  the  sum  of  the  several  weights  or  the  number 
of  single  measures.  Subtracting  each  M  from  z  gives  the 
residuals  in  the  column  v\  next  from  a  table  of  squares  the 
numbers  in  the  column  v*  are  found,  and  multiplying  each  of 
these  by  the  corresponding  weight  gives  the  quantities  pv* 
whose  sum  is  62.95.  Then,  since  n  is  6,  the  probable  error 
of  an  observation  whose  weight  is  unity  is  found  from  the 
formula  to  be  rl  =  2". 39  and  that  of  the  weighted  mean  to 
be  r  =  o".52.  Hence  the  final  value  of  the  angle  may  be 
written  z  —  32°  of  i8".l6  ±  o".52,  which  indicates  a  high 
degree  of  precision. 


10.        PROBABLE   ERRORS   FOR   INDIRECT   OBSERVATIONS.         29 

Prob.  9.  Four  measurements  of  a  base  line  give  the  results 
922.220  feet,  922.197  feet,  922.221  feet,  and  922.217  feet.  Compute 
the  probable  error  of  the  most  probable  value. 


10.  PROBABLE  ERRORS  FOR  INDIRECT  OBSERVATIONS. 

x- 

It  is  sometimes  required  to  find  the  probable  errors  of  the 
observed  quantities  Ml ,  M^ ,  .  .  .  Mn  ,  and  the  probable  errors 
of  the  quantities  x,  y,  2,  etc.,  whose  values  have  been 
obtained  by  the  methods  of  Arts.  7  and  8.  These  may  be 
found  by  first  deducing  the  probable  error  of  an  observation 
of  the  weight  unity  and  then  dividing  this  by  the  weights 
A  »  A  »  •  •  •  pn  and  px  ,  py  ,  pz ,  etc.  If  n  is  the  number  of 
observations,  q  the  number  of  unknown  quantities,  and  "2pv* 
the  sum  of  the  weighted  squares  of  the  residuals,  then,  as 
shown  in  treatises  on  the  Method  of  Least  Squares, 


(10) 


is  the  formula  for  the  probable  error  of  an  observation  of  the 
weight  unity,  and 

r  _  12-  r    -  Jl 

I/A  *# 

are  the  probable  errors  of  Ml  and  of  x  respectively. 

The  weights  /, ,  /,,.../„  are  known,  but  the  weights 
px ,  py,  etc.,  are  to  be  derived  by  preserving  the  absolute 
terms  of  the  normal  equations  in  literal  form  during  the  solu- 
tion. Then  the  weight  of  any  unknown  quantity  is  the 
reciprocal  of  the  coefficient  of  the  absolute  term  which  belongs 
to  the  normal  equation  for  that  unknown  quantity.  For 
instance,  take  the  normal  equations 

1*  -  y  -  z  =  A , 

-  *+  ar  —  *  =  A, 

—  x  —  y  +  2z  =  A . 


30  THE   METHOD   OF   LEAST   SQUARES. 

The  solution  of  these  by  any  method  gives 


Hence  the  weight  of  x  is  |,  that  of  y  is  J-,  and  that  of  z  is  I. 
If  it  be  only  desired  to  find  the  weight  of  x,  the  terms  D^ 
and  D^  need  not  be  retained  in  the  computation;  if  only  to 
find  the  weight  of  zt  the  terms  D^  and  D^  can  be  omitted. 

As  a  numerical  example  the  observation  equations  given 
at  the  beginning  of  Art.  5  may  again  be  considered.  These 
may  be  written,  if  x,  yy  and  z  denote  the  most  probable 
elevations, 

x  —  10.35  =  vl9 

y  —  x  —     7.25  =  vt, 

y  -   17.63  =  vt, 

y  —  z  —     9.  10  =  vtt 

x  —  z  -       1.94  —  vtt 

in  which  vt  ,  vt  ,  etc.,  are  the  residual  errors.  Now  in  Art.  6 
the  most  probable  values  were  derived, 

x  =  10.37,     y  =  17.61,      and     z  =  8.47  feet, 
and  substituting  these,  the  residuals  are  found  to  be 
vt=  +0.02,   z/,  =  —  o.oi,  v3=—  0.02,  v^=  +o-°4»  v§=—  0.04. 

Now,  as  the  weights  are  equal,  ~2pv*  becomes  2v*,  and  its 
value  is  2v*  =  o.o  041.  Then,  since  n  is  5  and  q  is  3, 


/O.OO4I 
A  / —  =  o.< 

V   5-3 


r,  =  0.6745^7      _^   =  0.031  feet, 

which  is  the  probable  error  of  a  single  observation. y1  (By  the 
method  above  explained  it  will  be  found  that  the  weight  of 
x  is  1.6,  whence  its  probable  error  is 

0.031  ~\ 

,  =   — =  =  0.024  feet, 
r  1. 6 


II.  PROBABLE    ERRORS   OF   COMPUTED    VALUES.  31 

and  in  a  similar  manner  the  probable  errors  of  y  and  z  are 
0024  feet  and  0.031    feet.    .The  final  adjusted  values  may 
then  be  written 
x  =  10.37  ±  °-02>  "  y  —  I'fkl  ±  0.02,      z  =  8.47  ±  0.03. 

Prob.  10.   Four  measurements  give  the  observation  equations 

-|-  x  =  12.27,     witn  weight  2, 

—  x  +  y  =     1.04,     with  weight  2, 

—  y  -|-  z  —     3.30,     with  weight  i, 

z  —  16.67,     with  weight  i. 

Find  the  most  probable  values  of  #,  y,  and  z,  their  weights  and  their 
probable  errors. 

11.  PROBABLE  ERRORS  OF  COMPUTED  VALUES. 

The  determination  of  the  precision  of  quantities  which  are 
computed  from  observed  quantities  is  now  to  be  discussed. 
For  instance,  the  area  of  a  field  is  computed  from  its  sides 
and  angles;  when  the  most  probable  values  of  these  have  been 
found  by  measurement,  the  most  probable  value  of  the  area 
is  computed  by  the  rules  of  geometry,  and  the  precision  of 
that  area  will  depend  upon  the  precision  of  the  observed 
quantities. 

Let  2^  and  2t  be  two  adjusted  values  whose  probable  errors 
are  rl  and  ra;  it  is  required  to  find  the  probable  error  r  of  the 
sum  2  =  2^  -f-  zy  If  z//,  z;,",  etc.,  be  residual  errors  for  zl 
and  z>3',  v9",  etc.,  be  residual  errors  for  #„  ,  then  the  corre- 
sponding errors  for  z  are  v'  =  z>/  -f-  z>/,  v"  =  v"  +  v"t  etc. 
Squaring  each  of  these  and  adding  the  results  gives 


and  for  a  large  number  of  errors  2vj>t  is  zero,  since  each 
product  vjj^  is  as  likely  to  be  positive  as  negative.  Now 
.2V,  2v*,  and  2v^  are  proportional  to  r\  r*,  and  r,a  as  seen 
by  (9),  and  accordingly 


32  THE    METHOD    OF   LEAST   SQUARES.  I. 

gives  the  probable  error  of  the  sum  £,  -|~  £a.  In  like  manner 
it  may  be  shown  that  the  probable  error  of  the  difference 
zl  —  #,  is  also  give.n  by  Vr*  -f-  ra*.  Further,  if  z  =  zt  ±  z% 
±  .  .  .  ±  s*  ,  then 

r*  =  rl*  +  rS  +  .  .  .  +  rj  (11) 

determines  the  probable  error  of  £.  For  example,  if  a  base 
line  be  measured  in  three  parts  giving  250.33  ±  0.05,  461.29 
±  0.07,  and  732.40  ±  o.  10  feet,  then  r  =  0.13  feet,  and  the 
total  length  may  be  written  1444.02  ±0.13  feet. 

If  x  be  an  observed  quantity  whose  probable  error  is  r, 
then  the  probable  error  of  ax  is  ar.  Thus,  if  the  diameter 
of  a  circle  be  observed  to  be  42  feet  2  inches  ±0.5  inches, 
the  circumference  is  132.47  i  0.13  feet. 

If  X  be  any  function  of  x,  then  the  error  dx  in  x  produces 
the  error  dX  in  X,  and  the  error  r  in  x  produces  the  error 

7  y 

r-j—  in  X.      For  example,  let  x  be  the  observed  diameter  of 

a  circle  and  r  its  probable  error;  then  X  =  \nx*  is  its  area, 
and  dX  =  ^nx-dx,  whence  the  probable  error  of  X  is  T'^TTX. 
Thus,  if  x  is  42  feet  2  inches  ±0.5  inches,  the  area  is 
1396.46  ±  2.76  square  feet. 

Lastly,  let  X  be  any  function  of  the  independently 
observed  quantities  x,  y,  z,  etc.,  and  let  it  be  required  to  find 
the  probable  error  of  X  from  the  probable  errors  rl  ,  r,  ,  r  ,  , 
etc.,  of  the  observed  quantities.  If  the  measurements  are 
made  with  precision,  so  that  the  probable  errors  are  small,  it 
can  be  shown  that 


determines  the  probable  error  of  X.  For  example,  let  x  and 
y  be  the  sides  of  a  rectangular  field  and  X  =  xy  its  area. 
Then  the  probable  error  rl  in  x  gives  the  probable  error  r^y 
in  X,  and  the  probable  error  r,  in  y  gives  the  probably  error 


12.  CRITICAL    REMARKS.  3$ 

rjc  in  X,  so  that  (r,^)2  -f-  (r^x)*  is  the  square  of  the  resulting 
probable  error  of  X.  Ttyus,  if  x  —  50.00  ±  o.oi  feet  and 
y  —  200  ±  O.O2  feet,  the  ar§a  is  10000  ±  2.24  square  feet. 

Formula  (ii)/  will  be  frequently  used  in  the  following 
pages,  it  being  a  general  rule  that  includes  all  cases.  As 
another  illustration  let  A  and  B  be  two  points  whose  hori- 
zontal distance  apart  is  /,  and  let  6  be  the  vertical  angle  of 
elevation  of  B  above  A  ;  let  rl  be  the  probable  error  of  /,  and 
ra  the  probable  error  of  0.  The  height  of  B  above  A  is  given 
by  X  =  /  tan  B,  and,  by  the  application  of  the  formula, 
regarding  h  as  x  and  6  as  y,  there  results 

r2  =  (r,  tan  fff  +  (rt//cos'  ff)\ 

If  /  =  1035.2  ±  1.3  feet,  and  B  =  3°  10'  ±  02',  then 
rl  =  1.3  feet,  but,  to  make  the  computation,  r2  must  be 
expressed  in  the  same  unit  as  cos2  0,  that  is,  in  radians;  since 
3438  minutes  make  one  radian,  the  numerical  value  of  ra  is 
2/3438.  Then  are  found  r,  tan  6  =  0.072  feet,  (ra//cos"  6) 
=  0.604  feet>  whence  r  =  0.608  feet.  The  value  of  X 
being  1035.2  tan  3°  10'  =  57.27  feet,  this  may  be  written 
57-27  ±  0.61  feet,  and  thus  it  is  as  likely  as  not  that  the 
error  in  the  computed  height  is  less  than  0.61  feet.  Here  it 
is  seen  that  the  probable  error  in  the  small  vertical  angle 
produces  the  greater  part  of  the  probable  error  in  the  com- 
puted result. 

Prob.  ii.  In  a  plane  triangle  ABC  let  A  =  90°,  C  =  16°  04'  45" 
±  30",  and  a  =  6256.8  ±  0.7.  Compute  the  length  of  the  side  c 
and  its  probable  error. 

12.  CRITICAL  REMARKS. 

The  most  important  processes  for  the  adjustment  and 
comparison  of  observations  have  now  been  presented,  but 
the  brief  space  at  command  has  forbidden  extended  theoretic 
discussions  like  those  found  in  treatises  on  the  Method  of 
Least  Squares.  The  student  has  been  obliged  to  take  for 


UNIVERSITY  OF  CALIFORNIA 
DEPARTMENT  OF  CIVIL  ENGINEERS 


34  THE    METHOD   OF   LEAST   SQUARES.  I. 

granted  the  law  of  probability  of  error  and  the  formulas  for 
probable  errors,  but  otherwise  the  subject  has  been  developed 
in  logical  manner.  Legendre,  in  announcing  the  principle 
of  least  squares  in  1805,  gave  no  proof  of  its  correctness  or 
validity;  he  notes,  however,  that  this  principle  balances  the 
errors,  so  that  the  effect  of  the  extreme  ones  is  neutralized. 

In  mechanics  the  center  of  gravity  is  a  point  about  which 
all  the  particles  of  the  body  balance;  so  the  arithmetic  mean 
gives  a  value  about  which  all  the  errors  balance,  the  sum  of 
their  residuals  being  zero.  The  moment  of  inertia  of  a  body 
is  a  minimum  for  an  axis  passing  through  the  center  of 
gravity;  so  the  sum  of  the  squares  of  the  residual  errors  is 
to  be  made  a  minimum  in  order  to  find  the  most  probable 
values  of  an  observed  quantity.  The  radius  of  gyration  with 
respect  to  an  axis  through  the  center  of  gravity  bears  also  an 
analogy  to  the  probable  error.  Thus  the  Method  of  Least 
Squares  may  be  justified  by  the  mechanical  principles  of 
equilibrium. 

Numerous  applications  of  the  adjustment  of  observations 
will  be  given  in  the  following  Chapters,  and  a  simplification 
will  be  introduced  whereby  the  formation  of  normal  equations 
from  observation  equations  may  be  rendered  numerically 
easier.  A  treatment  of  conditioned  observations  by  the  use 
of  "  correlate  equations"  will  also  be  presented,  whereby 
the  work  of  computation  may  often  be  materially  shortened. 
As  measurements  become  more  and  more  precise  the  neces- 
sity for  rational  processes  of  adjustment  and  comparison 
becomes  greater  and  greater.  In  physics,  astronomy, 
geodesy,  and  wherever  precise  observations  are  taken,  the 
Method  of  Least  Squares  is  now  universally  used,  and  there 
is  little  doubt  but  that  in  future  years  all  books  on  surveying 
will  treat  more  or  less  of  its  principles  and  processes. 

A  list  of  writings  on  errors  of  observations  and  on  the 
Method  of  Least  Squares  from  1722  to  1876  will  be  found  in 
Transactions  of  the  Connecticut  Academy,  1877,  vol.  IV,  pp. 


12.  CRITICAL   REMARKS.  35 

151-222.  Many  of  these,  together  with  others  from  1877  to 
1888,  are  given  in  Gore's  ,Bfbliography  of  Geodesy,  published 
in  Report  of  the  U.  S.  Coajjt  and  Geodetic  Survey  for  1887, 
pp.  313-512. 

Prob.  12.  A  base  line  was  measured  in  three  parts,  the  values 
found  for  these  being  126.74,  219.18,  and  270.40  meters.  The  total 
length  was  then  measured  and  found  to  be  616.39  meters.  Find  the 
adjusted  length  of  the  base,  the  weights  of  the  four  observations 
being  17,  9,  8,  and  3. 

Prob.  i2a.  A  plane  triangle  has  the  angle  A  measured  ten  times, 
v7  measured  five  times,  and  C  measured  once.  The  sum  of  the 
iliree  mean  values  is  found  to  differ  d  seconds  from  180°.  How 
should  this  d  be  divided  among  the  three  angles  ? 

Prob.  \2b.  Solve  the  following  normal  equations,  and  find  the 
weights  of  the  values  of  x  and  w\ 

2x  —  y  —    z  —     w  =  +  0.53, 

—  x  +  4y            —    w  =  —  0.27, 

—  x  +  32:  +    w  =  -  0.50, 

—  *  —  y  +    *  +  5^  ^  +  0<I9- 


36  PRECISE  PLANE  TRIANGULATION.  II. 


CHAPTER   II. 
PRECISE    PLANE    TRIANGULATION. 

13.  COORDINATES  AND  AZIMUTHS. 

Plane  surveying  is  that  which  covers  an  area  so  small  that 
it  is  unnecessary  to  take  into  account  the  curvature  of  the 
earth's  surface.  Surveys  of  cities,  townships,  harbors,  and 
mines  are  usually  of  this  character.  The  field  operations  of 
the  plane  triangulations  of  such  surveys  do  not  differ  in 
principle  from  those  of  geodetic  triangulation,  the  latter 
being  merely  more  precise. 

In  geography  the  position  of  a  point  on  the  earth's  surface 
is  located  by  its  angular  distance  north  or  south  of  the 
equator  and  by  its  angular  distance  east  or  west  of  the 
meridian  of  Greenwich,  these  coordinates  being  called  latitude 
and  longitude.  In  plane  surveying  two  straight  lines  are 
imagined  to  be  drawn  at  right  angles  to  each  other,  one 
coinciding  with  the  meridian,  and  these  constitute  a  system 
of  coordinate  axes  to  which  points  are  referred  by  rectangular 
coordinates.  The  linear  distance  of  a  point  east  or  west  of 
the  meridian  is  called  its  longitude,  and  the  linear  distance 
north  or  south  from  the  other  axis  is  called  its  latitude.  The 
coordinate  axes  are  rarely  laid  out  on  the  ground,  but  upon 
the  maps  they  are  drawn,  as  also  lines  parallel  to  them  at 
regular  distances  apart,  thus  forming  a  system  of  squares  by 
which  points  are  readily  located. 

The  azimuth  of  a  line  AB  is  the  angle  that  it  makes  with 
a  meridian  drawn  through  the  end  A.  Azimuths  are  usually 
measured  around  the  circle  from  o°  to  360°;  thus  if  the 


13- 


COORDINATES   AND   AZIMUTHS. 


37 


azimuth  of  AB  is  40°  the  azimuth  of  BA  is  220°  in  plane 
surveying.  » 

There  are  in  use  ^several  Systems  of  reckoning  coordinates 
and  azimuths.  The  one  most  commonly  used  in  plane  sur- 
veying has  the  latitudes  positive  when  measured  north  and 
negative  when  measured  south,  while  the  longitudes  are 
positive  toward  the  east  and  negative  toward  the  west.  In 
this  system  azimuths  are  reckoned  around  from  the  north 
through  the  east,  the  azimuth  of  north  being  O°,  that  of  east 
90°,  that  of  south  180°,  and  that  of  west  270°.  This  system 
is  used  in  the  Handbook  for  Surveyors. 

In  geodetic  surveying  in  America  latitudes  and  longitudes 
are  reckoned  as  in  geography,  north  latitude  being  positive 
and  south  latitude  negative,  while  west  longitude  is  positive 


f-i 
H-2 
•H 

pi 

/ 

-1 

/ 

k 

-3 

t3     4-2      -hi 


0      -1      -3     -3 


and  east  longitude  is  negative.  Here  the  azimuths  are 
reckoned  from  the  south  around  through  the  west,  the 
azimuth  of  south  being  o°,  that  of  west  being  90°,  that  of 
north  1 80°,  and  that  of  east  270°.  This  system  is  also  em- 
ployed for  the  linear  coordinates  of  plane  surveys  based  on 
geodetic  triangulations,  and  it  will  be  used  throughout  this 
volume.  For  a  city  survey  the  origin  may  be  taken  through 
the  tower  of  the  city  hall,  or,  if  it  is  desired  to  avoid  negative 
latitudes  and  longitudes,  it  may  be  taken  near  the  southeast 
corner  of  the  city.  The  size  of  the  squares  drawn  upon  the 
map  will  depend  upon  its  scale;  the  side  of  a  square  is 


38  PRECISE   PLANE   TRIANGULATION.  II. 

usually  taken  as  I  ooo  feet  or  meters,  or  10000  feet  or 
meters. 

In  thinking  of  the  azimuth  of  a  line  the  student  should 
imagine  himself  to  be  standing  at  the  end  which  is  first 
mentioned  in  its  name  and  to  be  looking  toward  the  other 
end;  then  he  should  imagine  a  meridian  drawn  through  that 
end  toward  the  south,  and  the  angular  deviation  of  these 
lines,  measured  as  above  described,  is  the  azimuth.  Thus, 
let  the  azimuth  of  A C  in  the  above  figure  be  115°,  then  the 
azimuth  of  CA,  determined  by  drawing  a  meridian  through 
C,  is  115°  +  180°  or  295°.  For  all  cases  in  plane  surveying 
the  back  azimuth  of  a  line  is  180°  greater  than  the  front 
azimuth,  because  the  meridians  are  parallel. 

When  several  lines  radiate  from  a  station  and  their  azimuths 
are  known  the  angle  between  any  two  lines  is  found  by  taking 
the  difference  of  their  azimuths.  Thus,  if  the  azimuth  of 
AB  is  35°  17'  04"  and  that  of  AC  is  120°  46'  19"  the  angle 
BAG  is  85°  29'  15".  Again,  for  the  lines  AB  and  AG  let 
the  azimuth  of  the  first  be  as  before  and  that  of  AG  be  320° 
10'  02",  then  the  angle  GAB  is  75°  06'  52" ';  here  360°  is  to 
be  added  to  the  azimuth  of  AB  before  subtracting  from  it 
the  azimuth  of  AG. 

Let  Pl  and  P^  be  two  points  in  such  a  plane  coordinate 
system,  Ll  and  Z,  their  latitudes,  Ml  and  M9  their  longitudes, 
/  the  length  of  the  line  joining  the  points,  and  Z  the  azimuth 
of  PtP9.  Then,  if  Llt  Ml ,  /,  and  Z  are  given,  the  coordinates 
of  Pt  are 

/,,  =  Ll  —  /  cosZ,     Mt  =  M. \  +  /  sinZ,  (13) 

which  are  always  correct  if  cosZ  and  sinZ  are  used  with 
their  proper  signs  according  to  the  value  of  Z.  For  example, 
let  A  =  +  20  148.3  feet,  M,  =  +  45933-7  feet,  /=  7789.5 
feet,  Z  =  205°  36'  07";  here  both  cosZ  and  sinZ  are  nega- 
tive, and  the  computation  gives  L^  =  -f-  27  173.0  feet  and 
MI  =  -f-  42  567.7  feet  for  the  coordinates  of  the  second  point. 


14.  MEASUREMENT   OF  ANGLES.  39 

A  triangulation  is  a  cheap  and  accurate  method  for  deter- 
mining the  coordinates  o(  stations.  The  stations  are  first 
located  on  the  ground  so  a§,  to  give  good-shaped  triangles, 
two  of  them  being  so  placed  as  to  form  a  base  line  whose 
length  can  be  measured  with  precision.  The  angles  of  all 
the  triangles  are  then  observed,  and  from  these  and  the  length 
of  the  base  the  lengths  of  all  the  sides  are  computed.  The 
azimuth  of  one  of  the  sides  is  determined  by  astronomical 
observations,  and  from  this  and  the  angles  the  azimuth  of 
each  side  is  known.  The  coordinates  of  one  of  the  stations 
being  assumed,  the  coordinates  of  all  other  stations  are  com- 
puted. Lastly,  the  lengths  and  azimuths  of  the  sides  and 
the  coordinates  of  the  stations  are  recorded  as  the  basis  for 
topographic  surveys,  and  the  coordinate  system  being  plotted 
the  stations  are  laid  down  in  their  correct  relative  positions. 

In  this  Chapter  those  operations  of  plane  triangulation  will 
be  discussed  which  depend  upon  the  measurement  of  hori- 
zontal angles.  Strictly  speaking  triangulation  includes  base- 
line and  azimuth  observations,  as  these  must  be  made  before 
the  angle  work  can  be  fully  computed.  It  will  be  con- 
venient, however,  to  first  discuss  the  angular  measurements 
and  their  adjustments,  leaving  the  base  lines  to  be  treated  in 
Chapter  III  and  the  azimuth  observations  in  Chapter  V. 

Prob.  13.  Let  the  latitude  and  longitude  of  Q  be +6  131.31 
meters  and  +  36  414.60  meters,  the  length  QN  be  12  454.02  meters, 
and  the  azimuth  of  QN  be  300°  06'  31".  Draw  the  figure  and 
compute  the  latitude  and  longitude  of  N. 

14.  MEASUREMENT  OF  ANGLES. 

Horizontal  angles  are  measured  either  with  a  direction 
instrument  or  with  a  repeating  instrument.  A  direction 
instrument  has  no  verniers,  but  the  readings  are  made  by 
several  micrometer  microscopes  placed  around  the  graduated 
circle.  Any  engineer's  transit  may  be  used  as  a  repeating 


4O  PRECISE   PLANE   TRIANGULATION.  II. 

instrument,  and  the  following  notes  will  treat  of  work  done 
with  these.  A  good  transit,  having  two  verniers  reading  to 
half-minutes,  can  easily  measure  horizontal  angles  with  a 
probable  error  of  one  second  if  proper  precautions  be  taken 
to  eliminate  systematic  and  accidental  errors. 

Errors  due  to  setting  the  transit  or  the  signals  in  the 
wrong  position  cannot  be  eliminated,  and  hence  great  care 
must  be  taken  that  they  are  centered  directly  over  the 
stations.  If  the  graduated  limb  be  not  horizontal  the 
measured  angles  will  be  always  too  large  and  hence  the  levels 
on  the  limb  must  be  kept  in  true  adjustment.  All  the 
adjustments  of  the  transit,  in  fact,  must  be  carefully  made 
and  preserved  in  order  to  secure  precise  work. 

Errors  due  to  inaccurate  setting  of  the  verniers,  as  also 
those  due  to  eccentricity  between  the  center  of  the  alidade 
and  the  center  of  the  limb,  may  be  eliminated  by  reading 
both  verniers  and  taking  the  mean.  Errors  due  to  collima- 
tion  and  to  a  difference  in  height  of  the  telescope  standards 
may  be  eliminated  by  taking  a  number  of  measures  with  the 
telescope  in  its  normal  or  direct  position  and  an  equal  number 
with  it  in  the  reverse  position.  Errors  due  to  inaccurate 
graduation  may  be  eliminated  by  taking  readings  on  different 
parts  of  the  circle.  Errors  due  to  pointing  and  to  clamping 
may  be  largely  eliminated  by  taking  one  half  of  the  measures 
from  left  to  right  and  the  other  half  from  right  to  left. 
Lastly,  in  order  to  eliminate  errors  due  to  atmospheric  influ- 
ences it  is  well  to  take  different  series  of  measurements  on 
different  days. 

The  following  form  of  field  notes  shows  four  sets  of 
measurements  of  an  angle  HOKy  each  set  having  three  repeti- 
tions. The  first  and  fourth  sets  are  taken  with  the  telescope 
in  the  direct  position,  the  second  and  third  with  it  reversed. 
The  first  and  second  sets  are  taken  by  pointing  first  at  H  and 
next  at  K,  the  third  and  fourth  are  taken  by  pointing  first  at 
K  and  next  at  H.  At  each  reading  both  verniers  are  noted. 


14. 


MEASUREMENT   OF   ANGLES. 


The  vernier  is  never  set  at  zero,  but  the  reading  before 
beginning  a  set  is  usually  made  to  differ  by  about  90°  from 
that  of  the  preceding  set  so  as  to  distribute  the  readings 
uniformly  over  the- circle.  **In  the  first  and  second  sets  the 
mean  final  reading  minus  the  mean  initial  reading  is  divided 
by  3  to  give  the  angle;  in  the  third  and  fourth  sets  the  mean 

FIELD    NOTES   OF    HORIZONTAL   ANGLE  HQK. 


i 

8. 

ii 
rt 

& 
o 

Reading. 

Angle. 

is 

3S 

o 

o 

Q 

A 

B 

Mean 

(/) 

fc 

H 

H 

3 

D 

10      02 

oo 

30 

15-0 

62  25  10.  o 

Angle  at  Station  O. 

K 

197      17 

30 

60 

45-0 

Sept.  31,  1895,  P.M. 

H 

3 

R 

IOO      II 

30 

30 

30.0 

62   25   07.5 

Brandis  Transit,  No.  716 

K 

287    26 

60 

45 

52.5 

John  Doe,  observer. 

R.  Roe,  recorder. 

K 

3 

R 

190    01 

30 

45 

37-5 

62   25   23.3 

Clear,  air  hazy. 

H 

2    45 

15 

40 

27-5 

K 

3 

D 

280    55 

45 

60 

52.5 

62   25    30.0 

Mean  of  4  sets 

H 

93     39 

00 

45 

22.5 

HOK=  62*25'  17".  7 

initial  reading  minus  the  mean  final  reading  is  divided  by  3. 
The  mean  of  the  four  values  of  the  angle  is  62°  25'  17". 7, 
which  is  its  most  probable  value  as  determined  by  these 
observations. 

In  repeating  angles  the  following  points  should  be  noted. 
The  transit  should  never  be  turned  upon  its  vertical  axis  by 
taking  hold  of  the  telescope  or  of  any  part  of  the  alidade. 
The  limb  should  never  be  clamped  when  the  verniers  are 
read.  The  observer  should  not  walk  around  the  transit  to 
read  the  verniers,  but  standing  where  the  light  is  favorable 
he  should  revolve  the  limb  so  as  to  bring  vernier  A  before 
him  and  then  vernier  B.  The  observer  should  not  allow  his 
knowledge  of  the  reading  of  vernier  A  to  influence  him  in 
taking  that  of  B.  Care  must  be  taken  to  turn  the  clamps 


PRECISE   PLANE   TRIANGULATION. 


II. 


slowly  and  not  too  tightly.  If  these  precautions  be  taken, 
and  if  the  observer  becomes  skilled  in  manipulation  and  close 
reading  of  the  verniers,  it  will  be  possible  to  obtain  the  value 
of  an  angle  to  a  high  degree  of  precision  with  a  transit  read- 
ing only  to  minutes. 

Four  sets  taken  in  the  manner  just  described  constitute  a 
series.  The  number  of  series  required  will  depend  upon  the 
precision  demanded  in  the  work.  If  it  be  required  to  render 
the  probable  erro'r  of  the  final  result  about  one  second,  it 
will  generally  be  necessary  to  take  about  6  or  8  series.  By 
taking  these  in  one  day  a  smaller  probable  error  may  be 
found  than  if  they  were  taken  on  two  or  more  different  days, 
but  the  final  result  will  really  be  more  precise  in  the  latter 
case  because  it  eliminates  numerous  errors  due  to  atmospheric 
influences, 

When  three  lines  meet  at  a  station  there  are  but  two  inde- 
pendent angles  to  be  observed,  when  four  lines  meet  there 
are  but  three  independent  angles,  and  in  general  n  lines  give 
n  —  I  independent  angles.  It  is,  however,  generally  best  to 


measure  all  the  angles  resulting  from  the  combination  of  the 
lines  two  by  two.  Thus  for  three  lines  OA,  OB,  and  OC, 
the  angles  A  OB,  AOC,  and  BOC  should  be  observed;  for 
four  lines  six  angles  should  be  measured,  namely,  AOB, 
AOC,  AOD,  BOC,  BOD,  and'  COD;  in  general  for  n  lines 
the  measurements  should  be  distributed  over  \n(n  —  i) 
angles.  If  about  5  series  are  thought  necessary  for  one  inde. 


15.  PROBABLE   ERRORS   AND    WEIGHTS   OF  ANGLES.  43 

pendent  angle,  then  for  four  independent  angles  20  series 
are  required,  but  if  these  are  distributed  over  the  ten  com- 
bined angles  then  only  2  series  need  be  taken  upon  each. 
The  adjustment  of  the  observed  values  by  the  Method  of 
Least  Squares  gives  finally  the  most  probable  values  of  all  the 
angles  at  the  station. 

Another  method,  which  is  used  by  many  observers,  is  to 
measure  each  of  the  n  angles  included  between  the  n  lines; 
thus  for  the  four  lines  in  the  middle  diagram  the  angles 
AOB,  BOC,  COD,  and  DO  A  would  be  measured.  The 
theoretic  sum  of  these  being  360  degrees,  the  observations 
are  then  to  be  adjusted  to  agree  with  this  condition  and  at 
the  same  time  render  the  sum  of  the  squares  of  the  residuals 
a  minimum  (Art.  21). 

In  writing  the  letters  designating  an  angle  it  is  desirable  to 
do  so  in*  the  order  of  azimuths,  that  is,  standing  at  the  vertex 
of  the  angle  the  letter  on  the  left-hand  line  should  be  men- 
tioned first.  Thus  AOB  means  the  angle  from  the  line  OA 
around  to  OB,  but  BOA  means  the  angle  from  OB  around  to 
OA  and  it  is,  of  course,  360°  minus  AOB.  By  this  method 
an  angle  is  estimated  in  the  same  direction  around  the  circle 
as  is  azimuth,  and  thus  uniformity  is  secured  and  ambiguity 
avoided. 

Prob.  14.  Show  that  the  probable  error  of  the  value  of  HOK 
found  from  the  above  field  notes  is  3".  6. 

15.  PROBABLE  ERRORS  AND  WEIGHTS  OF  ANGLES. 

In  the  field  note-book  the  observations  are  recorded  in  the 
order  in  which  they  are  made,  but  it  is  desirable  before  the 
occupation  of  a  station  is  concluded  that  the  results  for  each 
angle  should  be  arranged  in  an  abstract  and  the  probable 
error  be  computed.  Thus  the  observer  gains  a  clear  idea  of 
the  precision  of  each  angle  and  is  able  to  decide  whether 
additional  measures  are  necessary.  The  weights  of  the  final 


44 


PRECISE    PLANE   TRIANGULATION. 


II. 


means  are,    however,   usually  assigned   from   the   number  of 
repetitions  rather  than  by  the  probable  errors. 

The  following  is  an  abstract  of  the  observations  of  an  angle 
PNE  measured  on  the  precise  triangulation  around  Lehigh 
University  in  1898,  each  result  being  the  mean  of  four  sets 
taken  in  the  manner  shown  in  the  field  notes  of  Art.  15,  and 

ABSTRACT    OF   HORIZONTAL   ANGLES. 


Date. 

1898. 

No.   of 
Reps. 

Angle  PNE. 

V 

V* 

Remarks. 

Oct.  3 

12 

12°   15'    19".  3 

-  2.9 

8.4 

Buff  and  Berger 

Oct.  4 

12 

13   -8 

+  2.6, 

6.8 

Transit. 

Oct.  10 

12 

16  .3 

+  O.I 

o.o 

Each  series  taken  by 
a  different  observer. 

Oct.  II 

12 

21     .8 

-  5-4 

29.2 

Oct.  17 

12 

09   -5 

+  6.9 

47-6 

Oct.  18 

12 

17   .6 

—  1.2 

1.4 

ri  =  2".  90 

/  =  72 

12°    15'    i6"-4 

^z/2  =  93-4   | 

•  j 

r   —  i".i8 

each  being  taken  by  a  different  observer.  Here,  proceeding 
as  in  Art.  9,  the  arithmetic  mean  of  the  six  observations  gives 
12°  15'  1 6". 4  as  the  most  probable  value  of  the  angle  PNE. 
The  column  headed  v  gives  the  residuals  found  by  subtract- 
ing each  observation  from  that  mean,  and  then  the  sum  of 
their  squares  is  found  to  be  93.4.  From  (9)  the  probable 
error  of  a  single  result  is  computed  to  be  2 ".90  and  from  (9)' 
the  probable  error  of  the  mean  is  \" .  18,  which  shows  a  good 
degree  of  precision  considering  that  the  observers  were  not 
experienced  and  that  the  transit  reads  only  to  minutes. 

A  young  observer  is  usually  tempted,  after  having  com- 
puted the  mean  and  found  the  probable  errors,  to  reject  some 
of  the  observations  which  have  the  largest  residuals,  in  order 
thereby  to  apparently  increase  the  precision  of  the  results. 
This  temptation  must  be  resisted,  as  an  unwarranted  rejec- 
tion is  equivalent  to  a  dishonest  alteration  of  field  notes. 


15.  PROBABLE    ERRORS   AND    WEIGHTS    OF   ANGLES.  45 

There  are,  however,  two  cases  where  an  observed  value  may 
properly  be  rejected,  namely,  if  it  is  evidently  a  mistake,  as 
when  the  degrees  and  minutes  of  the  angle  are  wrong,  and  if 
a  remark  in  the  note-book  shows  it  was  taken  under  unfavor- 
able conditions.  Some  observers  allow  themselves  the 
liberty  of  rejecting  an  observation  when  its  residual  is  greater 
than  five  or  six  times  the  computed  probable  error  of  a  single 
observation.  There  are  some  reasons  in  favor  of  this  prac- 
tice, but  more  observations  than  one  should  never  be  rejected 
in  this  way. 

Although  the  weights  of  observations  are  inversely  propor- 
tional to  the  squares  of  their  probable  errors,  it  is  found  that 
it  is  better  and  more  convenient  to  give  weights  to  angles 
from  the  number  of  repetitions  or  series  which  produce  them 
rather  than  from  their  computed  probable  errors.  If  trie 
number  of  observations  were  large  in  each  case  the  two 
methods  might  closely  agree,  but  in  ordinary  practice  they 
do  not.  An  observer  of  much  skill  and  experience  may  be 
allowed  to  assign  weights  to  his  angles  with  regard  both  to 
the  number  of  repetitions  and  to  the  probable  errors,  but  in 
general  it  has  been  found  best  to  make  the  weights  closely 
proportional  to  the  number  of  repetitions  provided  the 
measurements  are  taken  under  the  same  conditions,  that  is, 
by  observers  and  instruments  of  equal  precision. 

To  illustrate  let  PNE  =  12°  is'  16^.4.  ±  i" .2  as  found  by 
6  series,  PNF  =  35°  of  42".$  ±  4^.8  as  found  from  4  series, 
and  ENF  =  22°  52'  24". o  ±  2" . 4  as  found  from  6  series. 
Here  the  probable  errors  indicate  that  the  precision  of  PNE 
is  much  greater  than  that  of  ENF,  but  in  making  the  adjust- 
ment it  is  best  to  take  their  weights  as  equal  since  each  has 
been  found  from  the  same  number  of  measures.  Thus  the 
weights  of  the  three  observations  should  be  taken  as  6,  4, 
and  6,  or  as  3,  2,  and  3  in  making  the  adjustment. 

Prob.  15.  Show  that  the  adjusted  values  of  the  above  observations 
are  PNE  =  12°  15'  i7".o,  PJVF=  35°  07'  4i".6,  and  ENF  = 

22°   $2'   24-".6. 


46  PRECISE  PLANE  TRIANGULATION.  II. 

16.  THE  STATION  ADJUSTMENT. 

When  several  angles  have  been  measured  at  a  station  they 
are  to  be  adjusted  by  the  methods  of  Arts.  6  and  7.  It  is 
here  only  necessary  to  give  additional  examples  and  to 
explain  an  abridgment  whereby  the  numerical  work  is  sim- 
plified. 

As  an  example  involving  equal  weights  let  the  data  be  the 
same  as  on  page  18,  the  five  observation  equations  being 

x  =    55°  57'  58".68, 

=    48    49    13   .64, 
=  104   47    12   .66, 

^  +*=    54    38    15   -53. 

+  7  +  ^=  103    27   28   .99, 

in  which  x,  yy   and   2  represent  the    angles 
^Q  CNM,  MNQ.,  and   QNS.     Now  let  x, ,  y, , 

and  #,  be  the  most  probable  corrections  to 
the  measured  values  of  x,  7,  and  z>  so  that 

y  =  48    49    13   .64  +7,, 
z  =  54    38    15   .53  +  *t, 

represent  the  most  probable  values  of  the  quantities  x,  y, 
and  2.  Then  substituting  these  in  the  observation  equations 
the  latter  become 

+  x,  =        o".oo, 

+  /i  =0    .00, 

-}-  z,   =  O    .00, 

+  ^  4.^  =  -  o  .18. 
Next,  by  the  rule  of  Art.  6,  the  normal  equations  are 

•r'.  +  S/',  +    *,  =  +  0.16, 
y,  +  2at  =  —  o.i  8, 


l6.  THE    STATION   ADJUSTMENT.  47 

the  solution  of  which  gives  the  corrections 

;r,  =  +  o".  15,     7/=+o.04,     £,  =  —0.11, 
and  hence  the  most  -probable'  values  of  x,  y,  and  z  are 

*  -  55°  57'  58".83  -  67W, 
7  =  48   49    13   .68  =  MNQ, 

z  =  54   38    15   .42  =  g^VS, 

» 

and  from  these  by  addition  the  most  probable  values  of  the 
other  observed  angles  are 

x  +  /  =  104  47    12   .51  =  £Wg, 

j  -j-  z  =  103   27   29  .10  =  MNS. 

The  residuals  for  the  five  observation  equations,  found  by 
substituting  the  most  probable  values,  are  +  o".  15,  +  o".O4, 
—  0.15,  —o.i  I,  -fo.  11,  and  the  sum  of  their  squares  is 
0.0708,  which  is  smaller  than  can  be  obtained  by  any  other 
values  of  x,  y,  and  z.  From  (10)  the  probable  error  of  each 
of  the  given  observations  may  now  be  found  to  be  ±  o".  13. 

When  the  weights  are  unequal  the  method  of  Art.  7  is  to 
be  followed.  As  an  example,  let  the  following  be  three 
angles  measured  at  the  station  O: 

MO  A  =  46°  53'  29". 4  with  weight  4, 
MOC  =  135  27  II  .1  with  weight  9, 
AOC  —  88  33  41  .1  with  weight  2. 

Now  let  x  and  z  be  the  most  probable  values  of  any  two 
angles,  say  of  MO  A  and  MOC.  Then  the  observation  equa- 
tions are 

x  =     46°  53'  29".4,      weight  4, 

z  •=.  135    27    ii    .  i,      weight  9, 

z  —  x—     88    33    41    .1,      weight  2. 

Next  let  xl  and  z^  be  the  most  probable  corrections  to  the 
observed  values  of  x  and  2,  so  that 

*=    46°53/29".4  +  *i, 
z  =  135    27    ii    .1  +*„ 


48  PRECISE   PLANE   TRIANGULATION.  II. 

are  assumed  probable  values  of  x  and  z.     Let  these  be  sub- 
stituted in  the  observation  equations,  which  thus  reduce  to 

x^  =        o".oo,      weight  4, 

zl  =        o  .00,     weight  9, 

xl  —  zl  =  -f-    0.60,      weight  2. 

From  these  the  normal  equations  are  formed  ;   they  are 
6x,  —     2zl  =  +  1.20, 

—  2Xl-\~    IlZl    =    —    1.  2O, 

from  which  the  most  probable  corrections  are 


Finally,  the  adjusted  values  of  the  three  angles  are 

x  =    46°  53'  29".6  =  MOA, 

z  =  135    27   ii   .o  =  MOC, 

z  —  x  =     88    33  41   .4  =  AOC. 

Here  it  is  seen  that  the  observation  having  the  largest  weight 
receives  the  least  correction,  which  should  of  course  be  the 
case. 

It  is  well  to  note  that  the  numerical  part  of  the  assumed 
probable  values  may  be  anything  that  is  convenient;  thus  in 
the  last  example  46°  53'  oo".o  +  xl  might  be  taken  for  x, 
and  135°  27'  oo".o  +  zv  for  z,  then  the  values  for  xl  and  zl 
would  be  found  to  be  +  29".  6  and  +  u".o.  The  object  of 
introducing  xl  ,  yl  ,  and  zl  is,  however,  to  make  the  numbers 
in  the  right-hand  members  of  the  observation  and  normal 
equations  as  small  as  possible,  and  this  is  generally  secured 
by  taking  the  corrections  as  additions  to  observed  values. 

After  the  adjustment  is  made  the  azimuths  of  all  the  lines 
radiating  from  the  station  are  easily  found  by  simple  addition 
or  subtraction,  provided  the  azimuth  of  one  line  is  known. 
Thus  for  the  last  example  let  the  azimuth  of  OM  be  given 
as  279°  04'  i8".4,  then  the  azimuth  of  OA  is  325°  57'  48".O, 
and  the  azimuth  of  OC  is  54°  31'  29".4. 


I/.  ERRORS  IN   A  TRIANGLE.  49 

P"ob.  1 6.  Angles  measured  at  the  station  O  between  the  stations 
Z>,  K}  M,  and  C  gave  the  following  results: 

DDK—    66°  33'  43"-7o,     weight  2, 


KOM~= 

66 

14 

22 

.10, 

weight 

2, 

KOC  — 

108 

02 

29 

.62, 

weight 

i, 

MOC  = 

4i 

48 

07 

.02, 

weight 

2, 

COD  = 

185 

24 

47 

.65, 

weight 

2. 

State  the  observation  equations,  form  and  solve  the  normal  equa- 
tions, find  the  adjusted  angles,  and  show  that  the  adjusted  value  of 
COD  is  185°  24'  47". 41  with  a  probable  error  of  ±  o".2o. 

17.   ERRORS  IN  A  TRIANGLE. 

The  simplest  triangulation  is  a  single  triangle  in  which  one 
side  and  the  three  angles  are  measured  in  order  to  find  the 
lengths  of  the  other  sides.  The  precision  of  the  values  found 
for  these  sides  will  depend  upon  the  probable  error  of  the 
base  and  the  probable  errors  of  the  measured  angles.  The 
best  triangle  is  one  whose  angles  are  each  about  60  degrees, 
and  a  triangle  having  one  angle  less  than  30  degrees  is  not  a 
good  one,  as  will  now  be  shown. 

In  a  triangle  whose  sides  are  a,  b,  and  c,  let  the  angles 
A,  B,  and  C  and  the  side  a  be  obtained  by  measurement. 
The  sides  b  and  c  then  are 

s'm£  sinC 

b  =  a  —. —  A  ,          c  =  a  — — -r. 
sin^  smA 

Now  suppose  each  angle  to  have  a  probable  error  r\  then 
by  the  use  of  (n)'  the  probable  errors  in  b  and  c  are  found 
to  be 


rb  =  br  i/cotM  +  cot3^,      re  =  cr  Vcot'A  +  cot'C.     (17) 

If  A,  B,  or  C  is  a  small  angle  its  cotangent  is  large  and 
accordingly  rb  and  rc  may  be  great.  As  far  as  b  is  concerned 
the  smallest  value  of  rb  will  obtain  when  A  =  B,  and  as  far 
as  c  is  concerned  the  smallest  probable  error  results  when 
A  =  C't  or  the  three  angles  should  be  equal  and  each  be  60 


50  PRECISE   PLANE  TRIANGULATION.  II. 

degrees  in  order  that  the  precision  of  b  and  c  should  be  the 
same  and  each  be  as  small  as  possible. 

As  a  numerical  example  let  a  =  I  ooo  feet,  A  =  90°, 
B  =  10°,  C  =  80°,  and  let  the  probable  error  in  each  angle 
be  i'.  Here  by  computation  b=  173.65  feet,  c=  984.81 
feet,  and  then 

rt=  I73-65  X  5-6;  X  rt      rc  =  984.81  X  0.176  X  r. 

The  value  of  r  to  be  used  here  is  i'  expressed  in  radians,  or 
r  =  7r/(i8o  x  60)  =  0.000291.  Accordingly  the  probable 
error  of  b  is  0.29  feet  and  that  of  c  is  0.06  feet,  so  that  the 
computed  values  of  b  and  c  have  a  large  degree  of  uncertainty. 
It  will  be  noticed  that  b,  which  is  opposite  the  small  angle 
B,  is  liable  to  a  far  greater  error  than  is  c. 

For  a  second  example  take  the  triangle  in  which  a  =  i  ooo 
feet,  A  =  60°,  B  =  60°,  C  ~  60°,  and  let  the  probable  error 
in  each  angle  be  i'.  Here  b  =  i  ooo  feet,  c  —  i  ooo  feet, 
and  r  =  0.000291 ;  then  from  (17)  there  is  found  rb  =  rc  = 
0.24  feet,  so  that  the  probable  error  of  the  computed  side  b 
is  less  than  in  the  previous  case. 

The  uncertainty  of  a  line  is  the  ratio  of  its  probable  error 

•   A  1  '    A.   1  I  T"*1  "-*_1/*  *  1  1,1 


\  to  its  length.  J  Thus  in  the  first  numerical  example  the 
uncertainty  of  the  computed  value  of  b  is  0.29/173.65  =  -g-J-g- 
nearly,  and  that  of  the  computed  value  of  c  is  0.06/984.81 
=  rs--Jlnr  nearly.  In  the  second  example,  however,  the 
uncertainties  of  b  and  c  are  0.24/1  ooo  =  f-faf  nearly.  An 
uncertainty  of  -^-j^  is  greater  than  that  of  a  rough  linear 
measurement,  and  an  uncertainty  of  y^-g^  is  greater  than 
should  occur  in  the  lengths  of  the  lines  computed  in  precise 
triangulations.  In  primary  geodetic  triangulation  work  the 
uncertainty  of  the  computed  sides  of  the  triangles  is  usually 
about  ^nj-VoT'  ^us  tne  probable  error  in  a  line  30  ooo  meters 
long  would  be  o.  I  meters. 

From  formula  (17)  it  is  seen   that  the   uncertainties  in  the 
computed  values  of  b  and  c  are 


18.  THE   TRIANGLE   ADJUSTMENT.  5 1 


+  cot'^,     ue  =  r  Vcot'A  +  cot'C,      ( i ;)' 

and  hence  these  may  be  computed  without  knowing  the 
lengths  of  the  sides  b  and  &  If  the  probable  errors  of  A,  B, 
and  C  are  different,  let  them  be  represented  by  rl ,  r^ ,  and  rt; 
then  from  (I  i)', 


ub  =  Vr*  co?  A  +  r,1  cot*j9,     ue  =  Vr,1  cotM  +  r*  cota<^,    (i  7)" 

are  the  uncertainties  in  the  computed  lengths  of  b  and  c.  If 
the  base  a  has  a  probable  error  ra  ,  this  may  also  be  taken 
into  account  by  (n)',  and  it  will  be . found  that  the  term 
(rjti)2  must  be  added  to  the  other  terms  under  the  radical 
signs  in  formula  (17)". 

In  laying  out  a  triangulation  it  is  not  possible  to  locate  the 
stations  so  that  each  angle  may  be  approximately  60  degrees, 
but  it  should  be  kept  in  mind  that  this  is  the  best  possible  ar- 
rangement and  that  it  should  be  secured  whenever  feasible. 
Angles  less  than  30  degrees  should  not  be  used  except  in  un- 
usual cases,  or  when  the  distances  computed  from  them  are 
not  to  be  used  for  the  computation  of  other  distances. 

Prob.  17.  In  a  triangle  the  adjusted  values  of  the  observed  angles 
are  25°  18'  07",  64°  01'  26",  and  90°  40'  27",  each  having  a 
probable  error  of  i".  The  length  of  the  side  opposite  the  smallest 
angle  is  3  499.39  feet,  and  its  uncertainty  is  j-g-fanr.  Find  the  uncer- 
tainties in  the  computed  values  of  the  other  sides. 


18.  THE  TRIANGLE  ADJUSTMENT. 

When  the  three  angles  of  a  plane  triangle  have  been  meas- 
ured their  sum  should  equal  180  degrees,  but  as  this  is  rarely 
the  case  they  are  to  be  adjusted  so  as  to  fulfil  this  condition. 
This  is  readily  done  in  any  particular  case  by  the  methods  of 
Arts.  6,  7,  and  16,  but  more  convenient  rules  for  doing  it 
will  now  be  deduced. 

First,  let  the  three  observed  values  be  of  equal  weight,  and 


52  PRECISE    PLANE   TRIANGULATION.  II. 

let  these  be  A,  By  and  C.  Let  x  and  y  be  the  most  probable 
values  of  A  and  B\  then  the  observation  equations  are 

x  =  A,     y  =  B,      180°  —  x  --  y  =  C. 

Now  let  v^  and  v^  be  the  most  probable  corrections  to  be 
applied  to  A  and  B  in  order  to  give  the  most  probable  values 
of  x  and  jy,  or 

x  =  A  +  v, ,         y  =  B  +  vt. 

Substituting  these  in  the  observation  equations,  the  latter 
reduce  to 

vt  =  o,     vt  =  o,     ^4-^=1 80°  —  A  —  B  —  C. 

Letting  d  represent  the  small  quantity  180°  —  (A  +  B  +  C) 
the  normal  equations  are  found  to  be 

27'i  +  v*  =  d>          vi  +  2vi  =  d, 

whose  solution  gives  vl  =  \d  and  v9  =  \d,  which  are  the 
corrections  to  be  applied  to  A  and  B.  Then  the  correction 
to  be  applied  to  C  is  also  \d*  Hence  the  rule:  Subtract  the 
sum  of  the  angles  from  180°  and  apply  one  third  of  the  dis- 
crepancy to  each  of  the  measured  values.  For  instance,  if 
the  three  measured  angles  are  64°  12'  19" '.3,  80°  o/  47". o, 
and  35°  39'  55". 8,  their  sum  is  180°  oo'  02".!,  and  the  dis- 
crepancy d  is  —  02".!.  Then  o"./  is  to  be  subtracted  from 
each  angle,  giving  64°  12'  i8".6,  80°  07'  46". 3,  and  35°  39' 
5  5".  I  as  the  most  probable  values. 

Secondly,  let  the  three  observed  values  be  of  unequal 
weight.  Let  these  be  A  with  weight  /, ,  B  with  weight/,, 
and  C  with  weight  /,.  The  observation  equations  are  the 
same  as  before,  but  are  weighted,  namely, 

Vi  =  O,      with  weight  /, , 

?;a  =  o,      with  weight/,, 
z/,  -f-  v^  =  d,      with  weight  /3. 
From  these  the  normal  equations  are 

(P,  +  A)v>  +  A",  = 
A",  +  (A  -f  AK  - 


IQ.  TRIANGLE  COMPUTATIONS.  53 

whose  solution  gives  the  corrections  vl  and  v^  and  then  the 
correction  vt  is  d  —  z>,  —  v^  "   Accordingly  the  results  are 

d  *      d  d 


in  which,  for  abbreviation,  P  represents 1 \- —  •     These 

A      A      A 

formulas  show  that  the  corrections  are  inversely  as  the 
weights,  so  that  the  angle  having  the  smallest  weight  receives 
the  largest  correction.  For  example,  let  the  weights  of 
A,  B,  and  C  be  10,  5,  and  I ;  then  vl  =  ^d,  v^  =  T2g^,  and 
vt  =  ^%d,  so  that  the  correction  for  C  is  ten  times  that  for  B 
and  five  times  that  for  A. 

If  only  two  angles  of  a  triangle  are  measured  there  can  be 
no  adjustment  made.  If  A  and  B  are  given  by  observations 
these  are  the  most  probable  values  of  those  angles,  and  the 
most  probable  value  of  C  is  180°  —  A  —  B.  In  all  precise 
primary  work  the  third  angle  should  be  measured  as  a  check, 
as  also  to  show  the  precision  of  the  observations,  whenever 
it  is  practicable.  Spires  and  other  inaccessible  points  may, 
however,  be  used  as  stations  in  secondary  triangulation. 

Prob.  18.  The  observed  angles  of  a  triangle  are  74°  19'  i4"-3 
with  weight  3,  35°  10'  42^.6  with  weight  7,  and  70°  30'  O9".4  with 
weight  9.  Find  the  adjusted  values  of  the  angles. 

19.  TRIANGLE  COMPUTATIONS. 

The  computation  of  the  sides  of  a  triangle  is  a  simple 
matter,  one  side  having  been  measured  as  a  base  line  or  being 
known  from  preceding  computations.  The  theorem  used  is 
that  the  sides  are  proportional  to  the  sines  of  their  opposite 
angles;  thus  in  the  triangle  ABC  let  the  side  AB  be 
known,  then 

log  CA  =  log  AB  —  log  smC  -(-  log  sinB, 
log  CB  =  log  AB  —  log  s'mC  -f-  log  sinA. 


54 


PRECISE   PLANE   TRIANGULATION. 


II. 


In  making  these  computations  it  is  desirable  that  a  uniform 
method  should  be  followed,  and  the  following  form  for 
arranging  the  numerical  work  is  recommended,  it  being 
similar  to  that  used  by  the  U.  S.  Coast  and  Geodetic  Survey. 

COMPUTATION   OF  A    PLANE   TRIANGLE. 


Lines  and 
Stations. 

Distances  and  Angles. 

Logarithms. 

B      \ 

AB 

2753.53 

3.4398898 

C 

49°     04'     49".  28 

0.1216914 

\                   ^^^C 

A 

90      21      24  .66 

1.9999916 

\                             >/ 

B 

40      33      46  .06 

I.8I3IOII 

\y/ 

CB 

3643.95 

3.5615728 

A 

CA 

2369.64 

3.3746823 

| 

Here  the  stations  are  arranged  in  the  order  of  azimuth,  and 
that  is  placed  first  which  is  opposite  to  the  given  side,  the 
length  of  this  and  its  logarithm  being  put  on  the  top  line. 
Opposite  the  second  and  third  angles  are  written  their 
logarithmic  sines,  and  opposite  the  first  angle  the  arithmetical 
complement  of  its  logarithmic  sine.  Now,  to  find  the  log  of 
CB  the  logarithm  opposite  B  is  to  be  covered  with  a  lead- 
pencil  and  the  other  three  logarithms  added.  So  to  find  the 
log  of  CA  the  logarithm  opposite  A  is  to  be  covered  and  the 
other  three  logarithms  added.  Lastly,  the  distances  corre- 
sponding to  these  logarithms  are  taken  from  the  table. 

If  the  precision  of  angle  work  extends  to  seconds  or  tenths 
of  seconds,  as  it  does  on  primary  triangulation,  a  seven-place 
table  of  logarithms  will  be  needed.  Six-place  tables  are 
rarely  found  conveniently  arranged  for  rapid  and  accurate 
computation.  For  a  large  class  of  secondary  work  five-place 
tables  are  sufficiently  precise.  In  taking  a  log  sin  from  the 
tables  the  student  should  note  that  the  characteristics  9.  and 
8.  mean  1.  and  2.  and  should  write  them  in  the  latter  manner 
in  his  computations. 


19.  TRIANGLE   COMPUTATIONS.  55 

When  the  above  triangle  ABC  is  connected  with  a  coordi- 
nate system  the  azimuth  of  AB  is  known  from  previous 
computations.  Then,  fro&i  this  and  the  angles  A  and  B,  the 
azimuths  of  AB  and  BC  are  easily  found.  Let  the  azimuth 
of  AB  be  149°  42'  55".68;  then  that  of  BA  is  329°  42' 
5  5 ".68,  and  accordingly 

Azimuth  AC  =  azimuth  AB  +  angle  A  —  240°  04'  20". 34, 
Azimute  BC  =  azimuth  BA  —  angle  B  =  289    09   09   .62. 

As  a  check  on  these  azimuths  it  may  be  noted  that  the 
second  minus  the  first  should  be  equal  to  the  angle  C. 

The  next  computation  is  that  of  finding  the  coordinates  of 
C  from  those  of  A  and  B.  For  the  above  triangle  suppose 
that  the  coordinates  of  A  have  been  assumed  and  that  those 
of  B  have  been  computed  from  (13),  the  values  being 

Station.  A  B 

Latitude  10000.00  12377.76 

Longitude  8  ooo.oo  9  388.59 

and  let  it  be  required  to  compute  the  latitude  and  longitude 
of  C.  These  should  be  found  in  two  ways  by  the  formulas 
in  (13),  so  as  to  check  the  correctness  of  the  results,  and  the 
form  below  shows  how  the  numerical  work  may  be  arranged 
in  a  systematic  manner.  In  the  first  column  /  denotes  the 
length  of  AC  or  BC,  the  logarithm  of  the  former  being  put 
in  the  third  column  and  that  of  the  latter  in  the  fifth  column. 
Similarly  Z  denotes  the  azimuth  of  AC  or  BC  whose  values 
are  given  in  the  second  and  fourth  columns;  adjacent  to 
these  are  written  the  values  of  log  cosZ  and  log  sinZ,  1. 
being  written  instead  of  the  9.  in  the  tables.  Then  log  / 
added  to  log  cosZ  gives  log  /  cosZ,  and  log  /  added  to  log 
sinZ  gives  log  /  s'mZ.  The  values  of  /  cosZ  and  /  s'mZ  are 
next  taken  from  the  logarithmic  tables  and  placed  in  the  second 
and  fourth  columns.  Opposite  Ll  and  J/,  are  placed  the  lati- 
tudes and  longitudes  of  A  and  B,  and  the  values  of  /  cosZ 
and  /  s'mZ  are  added  to  or  subtracted  from  them  as  required 


PRECISE  PLANE  TRIANGULATION. 


II. 


by  the  signs  of  cosZ  and  sinZ.  It  will  be  better,  however, 
for  the  student  to  determine  whether  these  are  to  be  added 
or  subtracted  by  drawing  figures  at  the  top  of  the  table. 


COMPUTATION  OF  COORDINATES. 


C 

i  /c 

C 

\ 

COMPUTED 

\     s' 

COMPUTED 

FROM 

A\ 

FROM 

^^\^^ 

Symbols. 

4. 

\ 

B. 

^\ 

1 

J                         L 

Distances  and 
Azimuths. 

Logarithms. 

Distances  and 
Azimuths. 

Logarithms. 

, 

3.3746823 

3-56I5728 

log  c<  sZ  = 

1.6980292 

log  cosZ  = 

I.5I59884 

z 

240°  04'  20".  34 

289°   09'    09".  62 

log  sinZ  = 

1.9378462 

log  sinZ  = 

1.9752699 

/cosZ 

I  182.26 

3.0727H5 

I  195-53 

3.0775612 

/sinZ 

2053.66 

3.3125285 

3442.25 

3.5368427 

£1 

10000.00 

12377.76 

Mv 

8  ooo.oo 

9388.59 

lat.  of  C 

II  182.26 

II  182.23 

long.  off 

5946.34 

5946.24 

If  the  computations  be  correctly  made  the  two  values  of 
the  latitude  of  C  must  exactly  agree,  as  also  the  two  values 
of  the  longitude  of  C.  In  this  case  there  is  a  discrepancy  of 
0.03  in  the  latitudes  and  of  o.io  in  the  longitudes  and  hence 
the  numerical  work  must  be  revised  so  as  to  detect  and 
remove  the  errors  of  computation. 

Prob.  19.  Revise  all  the  computations  in  this  Article  and  find  the 
correct  values  of  the  coordinates  of  C.  Also  make  the  computations 
for  the  triangle  DEF,  in  which  F  =  95°  24'  oi".o,  E  =  54°  58'  o8".6, 
&  =  29°  37'  5°"-4>  I>E=6  584.20  feet,  lat.  D  =  -f  15  328.75  feet, 
long.  D  =  -f-  12  047.05  feet,  azimuth  DE  =  216°  17'  05". 6,  and 
determine  the  coordinates  of  E\  finding  lastly  the  coordinates  of  F 
in  two  ways. 


2O. 


TWO   CONNECTED   TRIANGLES. 


57 


20.  Two  CONNECTED  TRIANGLES. 

Let  two  triangles  ABC  and  CD  A  have  the  side  AC  in 
common  and  let  all  the  angles  be  measured,  the  observations 
being  as  follows  and  all  of  equal  weight: 

^,=45°  19'  07",         C.=  50°  19'  37", 
A,  =  48   07   15   ,         C,  =  37   46  50  , 


=  96 


06   15 
54   19 


y2    =  93    26  28   , 

B  =  81  33  18  , 
Here  it  is  seen  that  the  sum  of  Al  and  A,  Is  06"  less  than  A, 
that  the  sum  of  Cl  and  £72  is 
12"  greater  than.  C,  that  the 
sum  of  Al  ,  B,  and  £Ta  is  10" 
greater  than  1  80°,  and  that  the 
sum  of  ^4a  ,  C19  and  Z>  is  06" 
greater  than  180°.  It  is  re- 
quired to  find  the  most  prob- 
able values  of  the  angles  which  entirely  remove  these  dis- 
crepancies. 

The  number  of  observed  angles  is  eight,  but  these  are 
subject  to  the  four  conditions  just  mentioned,  and  accord- 
ingly there  are  really  but  8  —  4  =  4  independent  angles  to 
be  used  in  the  computation.  Take  Alt  A,,  Ci  ,  and  C9  as 
these  independent  angles  and  let  at  ,  tf,  ,  cl  ,  and  c^  be  the  most 
probable  corrections  to  be  applied  to  the  observed  values. 
The  observation  equations  then  are 

a,  =        oo",  cl  =        oo", 

#,  oo  ,  ct  =        oo  , 

,       «i  +  a*  =  +  06   ,  Cl  +  c,  =  --  12   , 

a,  +<:,=  —  10  ,     a,  +  c,  =  -  06  . 

From  these  the  normal  equations  are  formed  ;   they  are 


+ 


4-    Ci  +  3'.  = 


00    , 

18  , 

•22-T 


CIVIL  ENGINEERING 

U.  of  C. 
ASSttCUTION  LIBRARY 


=  48  o;  15  .1, 
=  93  26  23  .6, 

=  81  33  13  -8,. 

C,  =  50"  19'  31'VI, 

C,  =  37  46  45  -5, 
C  =  88  06  16  .6, 
./?  =  96  54  06  .0, 

58  PRECISE    PLANE    TRIANGULATION.  II. 

and  the  solution  of  these  gives 

a,  =  +  00".  i,    ^  =  +  01".  5,     ^,  =  -04".  5,    *,=  —  os".9, 

as  the  most  probable  values  of  the  corrections  to  the  four 
angles.     Then  from  the  geometric  conditions  the  corrections 
to  the  other  angles  are 
«  =  *,  +  *,-  06"  =  -  4".  4,     c  =  Cl  +  r,  +  12"  =  +  1.6, 

b  =  —  at  —  cl  —  10"  =  —4.2,     d  =  —  al  —  c^—o6"  =  —  3.0, 

and  applying  these  to  the  observed  values  they  become 


A 


B 

which  are  the  most  probable  values  of  the  angles  and  which 
at  the  same  time  satisfy  the  geometry  of  the  figure. 

When  the  observations  are  of  unequal  weights  these  are  to 
be  used  in  forming  the  normal  equations  from  the  observation 
equations.  If  one  or  more  angles  are  unmeasured  these  do 
not  appear  in  the  observation  equations  and  their  values  are 
to  be  derived  from  the  adjusted  results.  If  the  angles  A  and 
C  are  not  measured,  but  all  the  others  are,  then  the  only 
adjustment  required  is  that  of  each  triangle  by  the  method 
of  Art.  1  8. 

If  the  length  and  azimuth  of  AB  and  the  coordinates  of  A 
be  given,  the  lengths  and  azimuths  of  the  other  lines  of  the 
figure,  as  also  the  coordinates  of  B,  C,  and  D,  may  be  com- 
puted by  the  methods  of  Art.  19.  Thus  a  simple  triangula- 
tion  is  established.  When  more  than  two  triangles  are  con- 
nected the  station  adjustments  are  usually  made  first,  and 
afterwards  the  triangle  adjustments  ;  cases  of  this  kind  are 
discussed  in  Chapter  IX. 

Prob.  20.  In  the  above  figure  let  the  observed  values  be  as  given 
except  that  of  Dy  which  is  not  measured.  Find  the  adjusted  values 
of  all  the  angles. 


21.  DIRECT   OBSERVATIONS   WITH    ONE   CONDITION.  59 


21.  DIRECT  OBSERVATIONS  WITH  ONE  CONDITION. 

In  Art.  1  8  are  given  examples  where  direct  observations 
on  several  quantities  are  connected  by  a  single  conditional 
equation,  and  as  other  cases  are  to  be  discussed  in  future 
Articles  it  will  be  well  to  derive  a  general  method  of  procedure 
which  will  simplify  the  numerical  work.  Let  x  and  y  be  two 
quantities  whose  values  have  been  fo-und  by  observation, 
these  having  the  weights/,  and  /,.  Let  these  quantities  be 
connected  by  the  conditional  equation 


in  which  ql  and  q^  are  known  coefficients,  and  D  is  a  known 
quantity.  Let  z^  and  v^  be  the  most  probable  corrections  to 
the  observed  values  so  that  the  observation  equations  are 

v,  =  o,    weight/,;      ^  =  o,    weight/,, 
and  the  conditional  equation  reduces  to 

£W  +  4W  =  d. 

Now  let  the  value  of  one  of  these  corrections  be  found  from 
the  last  equation  and  be  substituted  in  the  observation  equa- 
tions, and  then  let  the  normal  equations  be  formed  and 
solved,  and  finally  let  the  other  correction  be  found  from  the 
conditional  equation.  The  results  will  be 

q,d  q^d 

*-$*      v*  =  f,r' 

in  which,  for  abbreviation,  the  letter  P  represents  the  quantity 

£+£ 

A  ^  A 

The  same  process  may  be  extended  to  any  number  of 
unknown  quantities  and  similar  formulas  result.  Thus  if 
z>,  =  o,  va  =  o,  .  .  .  vn  =  o,  with  weights  /,,/,,  .  .  .  /„  ,  and 
if  the  conditional  equation  is 


60  PRECISE   PLANE   TRIANGULATION.  II. 

then  let  P=  y  +  q-j  +  .  .  .  +  ~-t  and   the  most   probable 

values  are 

q^  d  q^  d  qn  d 


which  also  exactly  satisfy  the  conditional  equation.  Formula 
(21)  hence  gives  a  general  solution  of  this  important  case. 

As  a  numerical  example  let  there  be  measured  at  a  station 
O  the  three  angles  AOB  =  97°  18'  20"  with  weight  5, 
BOC  =  135°  20'  OS"  with  weight  3,  COA  =127°  21'  29"  with 
weight  6.  Let  x,  y,  and  2  be  the  most  probable  values,  then 
must  x  -\-  y  -f-  z  =  360°.  Take  vl  ,  z'a  ,  and  vs  as  the  correc- 
tions to  the  observations,  and  the  conditional  equation 
reduces  to  v^  +  ^  +  v*  =  +  06".  Here  ql  =  q^  =  q^  =  i 
and  d  =  6";  also  /A  =  5,  /a  —  3,  /3  =  6,  and  hence  P  —  0.7 
and  d/P  =i  +  8.57.  Accordingly  from  (21)  the  values  of  the 
corrections  are  vl  =  -\-  i"  '.7,  z>4  =  -f-  2^.9,  z/,  =  -|-  i/7.4,  so 
that  the  most  probable  values  of  the  three  angles  which  satisfy 
the  conditional  equation  are  97°  18'  21".  7,  135°  20'  07".  9, 
and  127°  21'  30".  4. 

The  above  is  the  simplest  application  of  the  method  of 
correlates  which  is  extensively  used  in  the  adjustment  of 
geodetic  triangulations;  further  examples  of  it  will  be  given 
in  Chapter  IX.  For  the  case  of  equal  weights  the/'s  dis- 
appear from  the  above  formulas  and  P  becomes  the  sum  of 
the  squares  of  the  ^'s.  For  instance,  if  the  three  observed 
angles  of  the  last  paragraph  be  of  equal  weight,  then 
P  =  i  -\-  i  -)-  i  =  3  and  hence  vl  =  v^  =  v^  =  \d,  a  result 
which  agrees  with  the  rule  established  in  Art.  18  ;  accordingly 
the  adjusted  values  are  found  to  be  19°  18'  22",  135°  20'  07", 
and  127°  21'  3iv,  the  sum  of  which  is  180°. 

Prob.  21.  The  five  interior  angles  of  a  pentagon,  as  found  by 
measurement,  are  80°  19',  120°  57',  107°  04',  141°  35',  and  90°  oo'. 
Compute  the  adjusted  angles,  taking  the  weight  of  the  last  value  as 
three  times  that  of  each  of  the  others. 


22.  INTERSECTIONS    ON   A   SECONDARY   STATION.  6l 


22.  INTERSECTIONS  >  ON  A  SECONDARY  STATION. 

& 
After  a  triangulafion  has  been  established  any  side  may  be 

used  as  a  base  from  which  to  locate  a  secondary  station  by 
means  of  two  measured  angles.  If,  however,  a  third  station 
is  also  used  another  computation  may  be  made,  and  in  general 
the  results  will  not  exactly  agree  with  the  first  one  owing  to 
errors  of  observation.  An  adjustment  is  hence  to  be  made 
in  order  to  obtain  the  most  probable  position  of  the  secondary 
station. 

Let  ABC  be  a  triangle  whose  angles  are  known,  it  being  a 
part  of  an  established  triangulation.  At  the  three  corners 
let  the  angles  Al  ,  Biy  and  Cl 
be  measured  m  order  to  locate 
a  secondary  station  5.  The 
lines  determined  by  these  angles 
do  not  in  general  meet  at  the 
same  point,  and  hence  the  ob- 
servations are  to  be  adjusted 
to  secure  this  result.  The  con- 

dition that  the  three  lines  shall  meet  in  S  is  established  by 
equating  the  expressions  for  the  length  of  one  as  found  from 
another  in  two  ways;  thus  let  BS  be  found,  first  by  the 
triangle  ABS  and  secondly  through  the  triangles  ASC  and 
BSC;  the  values  are 


sine, 
and  hence  the  conditional  equation  is 

sin^t  sin^  s'mCl  =  smA^  sin^3  sin(72, 

which  must  be  exactly  satisfied  by  the  most  probable  values 
to  be  found  for  A,  ,  B,  ,  and  Ct.  This  is  called  a  side  equa- 
tion because  it  expresses  the  necessary  relation  between  the 


62  PRECISE    PLANE  TRIANGULATION.  IL 

three  lines  or  sides  which  meet  at  5.     By  taking  the  logarithm 
of  each  member  it  becomes 

log  sin^4i  +  log  sin2?i  +  log  sinCi — logsin^a — log  sin^2 — logsinCa=o, 
which  is  the  form  for  practical  numerical  work. 

As  an  example  let  the  given  angles  of  the  triangle  be 
A  =  83°  39'  01",  B  =  57°  19'  42",  and  C=  39°  01'  if. 
Let  the  three  angles,  as  measured  to  locate  5,  be  Al  =  41° 
05'  10"  with  weight  2,  B,  —  30°  15'  12"  with  weight  3,  and 
£\  =  1 8°  46'  of  with  weight  I  ;  it  is  required  to  adjust  these 
so  that  the  three  lines  may  meet  in  5  and  so  that  the  values 
found  may  be  the  most  probable. 

Let  a" ,  b" ',  and  c"  be  the  corrections  expressed  in  sec- 
onds to  be  added  to  the  observed  values  Al ,  Bl  ,  and  C,. 
Then  41°  05'  10"  +  a"  is  to  be  substituted  for  Al  in  the 
above  conditional  equation  and  similarly  for  B^  and  C}.  Now 
log  sin  (A,  +*")  =  log  sin  A,  +  0"-diff.  i"',  where  diff.  i' 
is  the  tabular  difference  for  one  second  corresponding  to 
the  value  of  A^\  thus  log  sin  (41°  05'  10"  +  a")  is  1.81769 
-\-  o.2^a" ',  where  0.23  is  in  units  of  the  fifth  decimal  place 
of  the  logarithm.  In  this  manner  the  following  tabulation  is 
made  : 

Observed  Angles.  Log.  Sines. 

A,  =  41°  05'  10"      1.81769  +  0.230" 

^   =   30      15      12  1.70228  -f  0.35^" 

Cl  =  18  46  07    1.50751  -{-  o.6i<r" 


1.02748  +  0.230"  -f  0.35^"  +  0.61^ 
A^  =  42  33  51    1.83021—0.220" 

^a  z=  27   04  30      1.65816  —  0.40^" 

(7,  =  20  15  10    1.53928  —  0.58^" 


1.02765  —  0.220"  —  0.40^"  —  0.58*:" 

the  values  of  A> ,  B^ ,  and  C9  being   those  found  by  subtract- 
ing Alt  BI,  and   (7,  from  A,  B,  and  C\  and  their  corrections 
being  the  negatives  of  a",   b" ,   and  c" .      Equating  the  two 
members  of  the  conditional  equation,  it  reduces  to 
0.450"  +  0.75^"  +  \.\gc"  =  17, 


22.  INTERSECTIONS   ON   A   SECONDARY   STATION.  63 

while  the  observation  equations  are  a"  =  o,  b"  =  o,  and 
c"  =  o,  whose  weights  are  2,  3,  and  i,  respectively. 

By  the  method  of  Art.  21   are  now  found  £,'//,  =  o.  101, 
q?/p^  —  0.187,   ?,V/3  —  1.440,   and    P  =  1.728,   whence  d/P 

—  _f_  9.83.       Then  *"   =  0.225   X    9.83    =   +   2//.2,    b"   = 
-f-  2". 4,  £•"  =  +  l  i"-8,  and  finally  the  logarithmic  corrections 
are  o.2^a"  =  -f-  i ,   0.358"  =  +  I,    etc.       Accordingly,    the 
most  probable  values  of  the  angles  and  of  their  logarithmic 
sines  are  found  to  be 

Adjusted  Angles.  Log.  Sines. 

Al  =  4i°05'  12"  1.81770 

-#1  —  3°    I5    J4  1.70229 

C}  —  18    46    19  1.50758 

1.02757 

^,  =  42  33  49  1.83021 
B^  —  27  04  28  1-65815 
C,  =  20  14  58  1-53921 

1.02757 

and  these  satisfy  the  geometric  conditions  of  the  figure  as 
closely  as  can  be  done  by  the  use  of  five-place  logarithms. 
From  these  angles  and  the  ^iven  lengths  of  AB,  BC,  and  CA 
the  distances  AS,  J3S,  and  CS  may  now  be  computed. 

The  above  method  also  applies  when  the  point  5  is  without 
the  given  triangle.  Thus,  if  j> 
be  situated  as  shown  in  the 
figure,  the  above  notation  can 
be  used  by  making  BAS  =  At  , 
CAS  =  A, ,  SCB  =  C, ,  and  SCA 

—  Cr     If   the   three   points  A, 
B,    and     C    fall    in    the    same 
straight    line,  the   method  fails, 

as  then  the  conditional  side  equation  is  satisfied  identically; 
in  this  case  the  distances  AB  and  BC  are  known  and  a  differ- 
ent side  equation  aiises  which  involves  these  lengths. 


64 


PRECISE    PLANE   TRIANGULATION. 


II. 


If  in  the  last  figure  there  be  given  the  distances  AB  and 
BC  and  the  angle  B,  and  if  Al ,  Blt  and  Ct  be  observed,  the 
condition  that  the  lines  AS,  BS,  and  CS  shall  meet  in  one 
point  is 

AB .  sinA,  sm(^  +  Q  =  BC .  sinC,  sin^,  +  B,\ 

which  may  be  used  in  a  manner  similar  to  that  of  the  above 
example.  Thus  let  there  be  given  AB  =  i  067.950  meters, 
BC=  883.839  meters,  B  =  135°  50'  51". 6,  and  let  there  be 
observed  ^,  =  75°  56'  oo".5,  £,  =  68°  34'  15". 2,  and 
Ct  =  81°  06'  35". o,  all  of  equal  weight.  Then  by  a  similar 
process  it  will  be  found  that  the  adjusted  values  of  these 
angles  are  A,  =  75°  56'  08". 8,  B,  =  68°  34'  08". 6,  and 
C*  =  81°  06'  37//.o,  and  that  the  two  values  of  BC,  computed 
from  these,  are  equal. 

Prob.  22.  Let  FG  and  £Zf  be  two  parts  of  a  straight  line,  each 
800  feet  long.  At  F,  G,  and  H  are  measured  the  angles  which 
lines  from  a  station  6"  make  with  the  base,  namely,  SFG  —  40°  i2f ', 
1?GS=  92°  58',  and  GHS  =  43°  55'.  Compute  the  length  of  GS 
in  two  ways,  and,  if  they  are  not  equal,  find  the  most  probable 
values  of  the  angles  which  will  effect  an  agreement. 


23.  THE  THREE-POINT  PROBLEM. 

In  secondary  triangulation  the  position  of  a  station  5  is 
sometimes  determined   by  measuring   the  angles  S,  and  S9 


subtended  at  it  by  three  stations  A,  B,  and  C  whose  positions 
are  known.  It  is  well  to  measure  the  three  angles  at  5  and 
then  by  the  station  adjustment  find  the  most  probable  values 


23.  THE   THREE-POINT   PROBLEM.  65 

of  5,  and  5a.  The  data  of  the  three  known  points  give  the 
distances  AB  and  BC  which  will  be  called  a  and  b,  and  also 
the  angle  CBA  which  wilfrbe  called  B.  The  problem  is  to 
determine  the  distances  SA,  SB,  and  SC. 

These  distances  can  be  found  as  soon  as  the  angles  A  and 
C  are  known.  Since  the  sum  of  the  interior  angles  of  the 
quadrilateral  is  360  degrees, 

A+C=  360°  -  B  -  S,  -  S,  ; 

and  since  the  side  £>S  is  common  to  two  triangles,  the  expres- 
sions for  its  length  when  equated  give 

s'mA       b  sin^S", 


sinC       a  sinS/ 

Thus  two  equations  are  established  whose  solution  will  give 
A  and  C.  Let  A  -\-  C  =  2m  and  A  —  C  =  2«.  The  value 
of  m  is  known,  namely, 


m=  i8o°-i(^  +  51  +  5'i),  (23) 

and  that  of  «  is  to  be  found.      Let  V  be  such  an  angle  that 

a  sinS2 


then  since  A  =  m  -f-  n  and  C  =  m  —  n,  the  second  equation 
becomes 


which  is  readily  reduced  to  the  form     , 

tanw  =  tanw  cot(F+  45°),  (23)x/ 

from  which  n  is  computed.  The  solution  is  hence  made  by 
first  finding  m  from  (23),  secondly  finding  V  from  (23)', 
thirdly  finding  n  from  (23)",  and  lastly  the  value  of  A  is 
•wi  n  and  that  of  C  is  m  —  «. 


66  PRECISE    PLANE    TRIANGULATION.  II. 

As  a  numerical   example   let   the  following    be   the  given 
data  for  three  stations,  as  determined  by  triangulation: 


Line. 

Azimuth. 

Distance. 

Station. 

Latitude. 

Longitude. 

ID 

327°  06'  49" 

9  on.o  ft. 

7 

34  !04-2 

5258i.5 

DJ 

74    56    58 

5  794-5 

D 

26537.2 

47  688.9 

Jl 

184    25    52 

9  098.9 

J 

25  032.5 

53  284.5 

At  a  station  5,  within  the  triangle  IDJ,  there  are  measured 
the  angles  ISD  =  127°  47'  33",  7XS/ =  87°  &  18",  and 
/57  =  144°  34'  09".  It  is  required  to  compute  the  lengths 
and  azimuths  of  57,  SD,  5/,  and  also  the  coordinates  of  5. 

Let  station  /  correspond  to  A  and  station  D  to  C\  then 
drawing  a  figure  and  comparing  it  with  that  above,  the  data 
are  5,  =  144°  34'  09",  5,  =  87°  38'  18",  B  =  74°  5$'  58"  - 
4°  25'  52"  =  70°  31'  06",  a  =  9098.9  feet,  £  =  5794-5  feet. 
Next  A  +  C=S7°  1 6'  27"  =  2»i,  and  01  =  28°  38'  14". 
From  (23)'  log  tan  Fis  found,  whence  V  —  69°  43'  13",  and 
then  from  (23)"  log  tan  n  is  found,  whence  n  =  —  14°  06' 
42".  Accordingly  ^  =  14°  31'  32"  =  57/,  and  C"  =  42°  44' 
56"  =  JDS.  From  the  triangle  ISJ  are  computed  the  dis- 
tances 5/=56oo.6  feet  and  £/ =  3936.6  feet;  from  the 
triangle  JSD  are  found  SJ  =  3936.6  feet  and  SD  =  4417.3 
feet.  The  azimuth  of  SD  is  74°  56'  58"  +  42°  44'  55"  + 
180°  —  297°  41'  54",  and  that  of  57  is  169°  54'  20".  Lastly, 
the  lengths  of  57  and  SD  are  multiplied  by  the  sines  and 
cosines  of  their  azimuths,  giving  the  differences  of  latitude 
and  longitude,  which  being  added  to  or  subtracted  from  the 
latitudes  and  longitudes  of  7  and  D  furnish  the  coordinates 
of  5  in  two  ways.  The  latitude  of  5  is  found  to  be  28  590.4 
feet  and  its  longitude  51  600.0  feet. 

A  theoretic  ambiguity  is  found  in  the  above  solution,  since 
Fand  n  may  each  have  two  different  values  corresponding  to 
the  values  of  tan  Fand  tan  n.  This  may  be  removed  by 
always  taking  Fas  less  than  90°  and  positive,  and  then  tak- 
ing n  as  less  than  90°  but  making  it  positive  or  negative 
according  as  tan  n  is  positive  or  negative. 


24. 


GENERAL   CONSIDERATIONS. 


When  the  point  6"  in  the  above  figure  falls  upon  the  cir- 
cumference of  a  circle  passing  through  A,  B,  and  C,  the  solu- 
tion is  indeterminate,  as  should  be  the  case.  When  5  lies 
very  near  this  circumference  the  results  of  the  computation 
will  be  uncertain.  In  such  an  event  a  fourth  station  should 
be  used  in  the  field  work. 

When  more  than  three  stations  are  observed  from  5  there 
arises  the  Appoint  problem,  in  which  three  different  locations 
for  5  can  be  computed  by  taking  the  stations  three  at  a  time. 
In  this  case  a  process  of  adjustment  by  the  Method  of  Least 
Squares  is  to  be  followed  so  that  the  four  lines  may  intersect 
in  one  point.  This  process  will  not  be  developd  here,  as  it 
is  of  infrequent  application  and  the  numerical  work  is  lengthy. 

Prob.  23.  Make  the  computations  for  the  triangle  IDJiioio.  the 
above  data,  letting  station  /  correspond  to  A  and  station  J  to  C. 

24.  GENERAL  CONSIDERATIONS. 

A  series  of  connected  triangles  with  one  or  more  measured 
bases  may  be  called  a  triangle  net.  The  purpose  of  the 


triangulation  and  the  topography  of  the  country  will  deter- 
mine the  location  of  the  stations  and  the  size  of  the  triangles. 
A  chain  net  is  one  suitable  for  a  river  survey,  a  polygonal 
net  where  the  triangles  from  one  or  more  polygons  is  some- 


68  PRECISE   PLANE   TRIANGULATION.  II. 

times  used  for  a  city  survey,  and  a  net  composed  of  quadri- 
laterals each  formed  by  four  overlapping  triangles  is  often 
used  in  geodetic  work.  The  three  types  are,  however, 
frequently  combined  together,  single  triangles,  polygons,  or 
quadrilaterals  being  used  in  different  parts  of  the  same  net. 

A  chain  net  is  the  simplest  in  adjustment,  since  no  side 
equation  arises  if  there  be  but  one  base.  In  the  other  kinds 
there  will  be  one  side  equation  for  each  polygon  and  one  for 
each  quadrilateral,  but  increased  labor  in  computation  counts 
for  little  when  precision  is  demanded.  A  quadrilateral  is  a_ 
figure  securing  high  precision,  and  the  polygon  takes  almost 
rank  with  it,  since  the  side  equation  eliminates  accidental 
errors  that  otherwise  might  be  propagated  along  the  net. 

In  the  preceding  pages  only  an  introduction  to  the  methods 
of  adjustment  has  been  given.  The  subject,  however,  will 
be  continued  in  Chapter  IX,  where  cases  involving  more  than 
one  conditional  equation  will  be  discussed. 

It  may  have  been  noticed  in  the  use  of  the  side  equation 
in  the  preceding  Articles  that  the  smallest  angles  receive  the 
largest  corrections  if  the  weights  of  the  observations  are 
equal.  It  hence  appears  to  be  important  in  conducting  the 
field  work  to  measure  angles  less  than  30°  or  greater  than 
150°  with  a  higher  degree  of  precision  than  those  between 
30°  and  150°.  By  so  doing  the  weight  of  the  smaller  angle 
will  overbalance  the  error  due  to  the  large  tabular  difference 
in  its  sine,  and  the  corrections  will  be  more  uniformly  dis- 
tributed among  the  measured  values. 

Geodetic  triangulation  nets  differ  from  plane  ones  only  in 
the  greater  size  of  the  triangles  and  in  the  fact  that  the  sum 
of  the  angles  of  each  triangle  is  greater  than  180°.  All  the 
preceding  methods  are  hence  directly  applicable  in  geodetic 
work.  When,  however,  a  plane  net  is  extended  for  some 
distance  east  or  west  of  the  meridian  where  the  initial  azimuth 
was  determined,  the  computed  azimuths  become  less  or 


24. 


GENERAL   CONSIDERATIONS. 


greater  than  the  true  ones  owing  to  the  curvature  of  the 
earth.  In  geodetic  work  this  discrepancy  is  removed  by 
introducing  a  correction  whteh  renders  the  back  azimuth  of  a 
line  different  from  its  front  azimuth,  each  value  being  the 
angle  which  the  line  makes  with  a  meridian  drawn  through 
the  end  considered. 

When  the  plane  coordinates  of  two  stations  are  known  the 
length  and  azimuth  of  the  line  joining  them  is  readily  com- 
puted. Thus,  let  Ll  and  Z,  be  the  given  latitudes,  then  the 
latitude  difference  L^  —  Ll  is  known;  also  let  Mt  and  M^  be 
the  given  longitudes,  then  the  longitude  difference  M^  —  Mt 
is  known.  From  (13)  it  is  seen  that  the  azimuth  from  the 
first  point  to  the  second  is  found  by 


(24) 


and  the  distance  /  ma'y  then  be  computed  from 

j\/r n/T  T      _   T 

•i'J-n    ^^     •LrJ-  i  J-^»    ^^     J-^i 


1   = 


s'mZ 


or     /  =  — 


cosZ 


As  an  example,  let  the  latitudes  of  two  stations  F  and  G  be 
given  as  15420.72  feet  and  18  115.13  feet,  and  their  longi- 
tudes as  20  347.19  feet  and  14  739.08  feet;  here  the  latitude 
difference  is  -|-  2  694.41  feet  and  the  longitude  difference  is 


AZIMUTH  AND  DISTANCE  FOR  FG. 


Symbols. 

Distances  and 
Azimuth. 

Logarithms. 

Mt  -  J/i 

-f  5608.11 

3.7488165 

za-za 

—  2  694.41 

3.4304637 

z 

244°       20'       1  7".  I 

0.3183528 

tans 

. 

1.9549007 

siru 

1.6365480 

cosz 

/ 

622I.8O 

3-7939I58 
3-7939I57 

70  PRECISE   PLANE   TRIANGULATION.  II. 

—  5  608.  II  feet.  The  computation  may  be  arranged  in  the 
form  as  shown.  The  second  logarithm  subtracted  from  the 
first  gives  log  tanZ  and  then  Z  is  taken  from  the  table;  as 
tanZ  is  positive  64°  20'  ij" .1  is  the  azimuth  of  GF and  244° 
20'  if.  I  is  the  azimuth  of  PG.  Then  log  sinZ  and  log 
cos^  are  taken  out,  and  the  subtraction  of  these  from  the 
first  and  second  logarithms  gives  two  values  of  log  /  which 
must  agree  within  one  unit  of  the  last  decimal.  Lastly  /  is 
taken  from  the  table.  Thus  the  distance  and  azimuth 
between  two  stations  which  are  not  connected  by  a  side  of 
one  of  the  triangles  may  be  quickly  computed  in  a  plane 
system  of  coordinates. 

Prob.  24.  The  latitudes  of  two  stations  M  and  TV  are  12  900.21 
and  9  883.85  feet,  and  their  longitudes  are  27  333.16  and  35  640.93 
feet.  Compute  the  distance  and  azimuth  from  M  to  N* 

Prob.  240.  In  Art.  22  it  should  be  noted  that  the  signs  of  the 
corrections  to  a  log  sin  are  to  be  taken  as  negative  for  an  angle 
between  90°  and  180°.  What  is  the  correction  to  a  log  sin  for  an 
observed  value  Al  =  123*  15'  30"? 


25.  PRINCIPLES   AND    METHODS. 


CHAPTER  III. 
BASE    LIN  ES. 

25.  PRINCIPLES  AND  METHODS. 

The  principle  involved  in  the  measurement  of  a  base  line 
is  the  same  as  that  in  common  chaining,  the  unit  of  measure 
being  applied  successively  from  one  end  of  the  line  to  the 
other.  It  is  very  important  that  length  of  the  measuring 
unit  should  be  accurately  known  in  terms  of  the  standard 
linear  foot  or  meter,  for  otherwise  its  absolute  error  may  be 
multiplied  so  as  to  give  an  erroneous  length  for  the  base. 

As  the  measuring  bars  or  tapes  are  of  metal  they  expand 
or  contract  as  the  temperature  rises  or  falls  and  hence  the 
coefficient  of  expansion  of  the  metal  must  be  known  in  order 
to  eliminate  errors  due  to  this  source.  Other  systematic 
errors,  like  those  due  to  pull  and  sag  in  a  tape  and  those  due 
to  the  inclination  of  the  base  to  the  horizontal,  must  also  be 
eliminated  by  computation.  Accidental  errors  due  to  indefi- 
nite causes  still  remain  in  each  result  and,  in  order  that  the 
final  length  may  be  largely  free  from  these,  the  measurement 
must  be  repeated  several  times  and  their  mean  be  taken. 

Metallic  bars  from  10  to  20  feet  in  length  have  been 
extensively  used  for  base  measurements.  These  are  of  two 
classes,  end  measures  and  line  measures.  With  end  measures 
the  distance  between  the  extremities  of  the  ends  is  a  unit, 
and  measurement  is  made  by  contact,  one  bar  being  placed  in 
position  and  another  brought  into  line  so  that  the  ends  of  the 
two  touch  each  other;  these  ends  are  usually  rounded  to  a 
radius  equal  to  the  length  of  a  bar.  With  line  measures  but 


72  BASE   LINES.  III. 

one  bar  is  required,  the  distance  between  two  marks  engraved 
upon  its  upper  surface  being  a  unit;  a  microscope  being 
placed  on  a  movable  frame  over  one  mark,  the  bar  is  moved 
forward  until  the  other  mark  comes  into  the  same  position, 
and  then  the  microscope  is  moved  forward  to  the  first  mark. 
In  each  case  the  number  of  bar-lengths  multiplied  by  the 
length  of  one  gives  the  length  of  the  base. 

End  measures  are  more  convenient  than  line  measures,  but 
are  generally  not  as  precise.  In  order  to  eliminate  effects  of 
temperature,  compound  bars  composed  of  metals  whose  rates 
of  expansion  are  different  have  been  devised  and  used;  in 
these  one  bar  expands  more  than  another,  so  that  by  the  use 
of  a  compensating  lever  the  distance  between  the  marks  or 
ends  is  supposed  to  remain  invariable. 

Since  1885  the  long  steel  tape  has  been  extensively  used 
in  the  measurement  of  base  lines,  and  has  been  shown  to  give 
results  of  a  high  degree  of  precision.  As  such  a  tape  can 
readily  be  bought  and  standardized,  as  its  use  involves  little 
expert  knowledge,  and  as  a  base  can  be  measured  with  it 
very  cheaply,  a  full  explanation  of  the  method  of  procedure 
will  be  given  in  later  Articles. 

Prob.  25.  Consult  Report  of  U.  S.  Coast  Survey  for  1897,  and  de- 
scribe the  duplex  base  apparatus,  and  ascertain  the  character  of  its 
work. 

26.  PROBABLE  ERROR  AND  UNCERTAINTY. 

As  a  line  is  measured  by  the  continued  application  of  a 
unit  of  measure  the  probable  error  in  a  result  found  for  its 
length  should  increase  with  that  length.  The  law  of  this 
increase  is  found  from  formula  (11);  thus  if  rl  be  the  prob- 
able error  of  the  unit  of  measure  and  /  be  the  length  of  the 
line,  the  probable  error  of  /  is 

r  =  r,Vl  (25) 

that    is,    the    probable    error    in    a    measurement    of   a    line 


26.  PROBABLE    ERROR   AND   UNCERTAINTY.  73 

increases  with  the  square  root  of  its  length.  Thus  if  two 
lines  are  measured  with  e,qual  care  and  the  second  is  four 
times  as  long  as  the  first,  t&e  probable  error  of  the  second 
measurement  is  twice  that  of  the  first  one. 

Since  weights  are  inversely  as  the  squares  of  probable  errors 
it  follows  that  the  weights  of  linear  measurements  made  with 
equal  care  are  inversely  as  the  lengths  of  the  lines.  Thus,  a 
measurement  of  I  ooo  feet  must  be  twice  repeated  and  the 
mean  of  the  results  be  taken  in  order  to  be  worth  as  much  as 
a  single  measurement  of  500  feet.  In  combining  linear 
measures,  therefore,  the  weights  of  observations  should  be 
taken  as  the  reciprocals  of  the  distances. 

The  most  convenient'way  to  find  the  value  of  rl  is  to  make 
duplicate  measures  of  lines  of  different  lengths.  Let  the 
lengths  of  the  lines  be  /t  ,/,,.../„,  the  differences  of  the 
duplicate  measures  be  dv ,  d^ ,  .  .  .  dn ,  and  n  be  the  number 
of  lines.  Then,  as  shown  in  treatises  on  the  Method  of 
Least  Squares,  the  probable  error  of  a  linear  unit  is 


r,  =  o.4769V  T*J-.  (26) 

For  example,  in  order  to  find  the  probable  error  of  measure- 
ment with  a  steel  tape  four  lines  were  measured  as  follows: 
7=427.34  854.21  1281.71  i  708.40  feet 

7=427.37  854.20  1281.74  i  708.33  feet 

d  =  —  0.03  -f-  o.oi  .  —  0-03  +0.07  feet 

p  =  0.00234         0.00117  0.00078  0.00059 

Here  the  weights  are  taken  as  the  reciprocals  of  the  lengths, 
since  the  weight  of  a  line  one  foot  long  is  taken  as  unity. 
Then  by  the  use  of  the  formula  the  probable  error  of  a 
measurement  one  foot  long  is  found  to  be  0.00058  feet,  and 
accordingly  that  of  one  100  feet  long  would  be  0.0058  feet. 
Of  course  a  larger  number  of  observations  than  four  is 
required  to  deduce  a  reliable  value  of  this  probable  error. 
The  uncertainty  in  the  length  of  a  line  is  expressed  by  the 


74  BASE   LINES.  III. 

ratio  of  its  probable  error  to  its  length  (Art.  26),  and  is  hence 
given  by  rj  Vly  where  r,  is  the  probable  error  of  a  line  one 
unit  in  length.  Accordingly,  if  a  certain  line  has  an  uncer- 
tainty of  T7nnnn>»  tne  uncertainty  of  a  line  four  times  as  long 
and  measured  in  the  same  manner  is  •$-$%-$-$.  It  thus  follows 
that  greater  errors  in  the  computed  sides  of  triangles  might 
result  from  a  long  base  than  from  a  shorter  one. 

Prob.  26.  Let  the  probable  error  of  measurement  with  a  steel 
tape  be  0,005  feet  for  100  feet.  A  square  city  lot  is  laid  out  with 
this  tape  so  as  to  contain  43  560  square  feet.  Compute  the  prob- 
able error  of  this  area. 

27.  BASES  AND  ANGLES. 

The  uncertainty  in  the  length  of  a  computed  side  of  a 
triangle  is  caused  by  a  combination  of  the  errors  in  the  base 
with  those  in  the  angles,  and  the  influence  of  the  angles  is 
usually  greater  than  that  of  the  base.  Let  the  base  a  in  the 
triangle  ABC  be  measured  with  a  probable  error  ra,  and  let  r 
be  the  probable  error  of  the  angle  measurements  expressed 
in  radians.  Then  by  (n), 


rb  =  b  V(ra/af  +  r2  cotM  +  r'  co?B 

is  the  probable  error  in  the  computed  value  of  b.  Now  in 
Art.  17  it  was  shown  that  the  best-shaped  triangle  is  an 
equilateral  one,  and  for  this  case  the  formula  gives 


ub  =     u  r 

as  the  uncertainty  in  the  computed  value  of  b.  Let  the 
probable  error  of  the  angle  observations  be  one  second  or 
0.000004848  radians.  Then,  if  the  base  were  without  error, 
the  uncertainty  in  b  would  be  ^-g-g-Vrnp  but  if  the  base  have 
also  an  uncertainty  of  ^jVmr  the  uncertainty  in  b  will  be 


It  is  not  easy  to  carry  on  a  triangulation  so  that  the  mean 
probable  error  of  the  adjusted  angles  shall  be  less  than  one 


27.  BASES   AND   ANGLES.  75 

second,  but  it  is  very  easy  to  measure  a  base  of  moderate 
length  so  that  its  uncertainty  shall  be  less  than  ^j^Voif  ^n 
geodetic  work  bases  have  been,  measured  with  an  uncertainty 
of  less  than  y  fruTTnnr*'  ^  thus  appears  that  even  in  the  best- 
proportioned  triangle  the  precision  of  the  base  measurement 
can  be  rendered  greater  than  that  of  the  angle  work.  The 
difficulty  of  finding  good  locations  for  bases  and  the  expense 
of  measuring  them  renders  it  customary,  however,  to  use  only 
one  or  two  in  a  triangulation  net  of  moderate  extent.  When 
the  sides  of  the  triangles  are  from  one  to  ten  miles  in  length 
a  base  line  about  a  mile  long  may  be  used.  Care  must  be 
taken  that  the  triangles  connecting  it  with  the  main  net  are 
well  proportioned,  no  angle  being  less  than  30  degrees.  The 
topography  of  the  country  will  determine  the  location  of  the 
stations  to  a  great  extent,  but  the  figures  show  two  methods 


of  gradually  increasing  the  lengths  of  the  sides  away  from  a 
measured  base  AB\  the  second  method  is  the  better  one. 

In  geodetic  work  bases  several  miles  in  length  have  been 
used.  For  example  a  base  of  the  U.  S.  Coast  and  Geodetic 
Survey  in  Massachusetts  is  nearly  lof  miles  long,  its  meas- 
urement occupying  three  months  in  1844.  The  final  result, 
reduced  to  the  ocean  level, -was  17326.376  ±  0.036  meters, 
giving  an  uncertainty  of  ^g-yVcro-  About  295  miles  north- 
easterly is  the  Epping  base,  and  230  miles  southwesterly  is 
the  Fire  Island  base,  which  were  also  measured  with  similar 
precision.  The  length  of  the  Massachusetts  base  as  computed 


76  BASE   LINES.  III. 

through  the  triangulation  from  the  Epping  base  was  found 
to  be  17  326.528  meters,  and  its  length  computed  from  the 
Fire  Island  base  was  found  to  be  17  326.445  meters.  The 
actual  uncertainties  between  the  measured  and  computed 
values  are  hence  TTTVorr  and  yy^Vinr  respectively,  the  effect 
of  the  errors  in  the  angles  being  four  times  that  of  the  base 
errors  in  the  first  case.  In  general  it  is  found  that  angle 
errors  do  not  increase  the  uncertainties  of  computed  lengths 
to  the  extent  that  might  be  inferred  from  the  preceding 
discussion,  and  this  is  probably  due  in  part  to  the  fact  that 
they  are  largely  eliminated  in  the  adjustment  of  the  triangu- 
lation. 

Prob.  27.  Four  measures  of  a  base  line  give  the  values  922,220, 
922.197,  922.221,  and  922.217  feet.  Show  that  the  uncertainty  of 
the  mean  of  these  measures  is  about 


28.  STANDARD  TAPES. 

A  long  steel  tape  is  the  most  convenient  apparatus  for 
measuring  the  base  line  of  a  river  or  city  survey,  and  it  has 
also  been  used  for  geodetic  bases  with  excellent  results.  It 
is  necessary  that  it  should  be  compared  with  a  standard,  and 
this  can  be  done  for  a  small  fee  by  the  Bureau  of  Weights 
and  Measures  at  Washington.  The  certificate  furnished  will 
state  the  error  of  its  length  for  a  certain  temperature  and 
pull,  or  it  will  state  that  it  is  correct  at  a  given  temperature 
and  pull.  The  coefficient  of  expansion,  or  the  relative 
change  in  length  for  i°  Fahrenheit,  should  also  be  stated,  in 
order  that  the  effect  of  temperature  may  be  eliminated.  The 
coefficient  of  stretch,  or  the  relative  change  in  length  for  one 
pound  of  pull,  must  also  be  known.  A  tape  thus  standard- 
ized becomes  itself  a  standard  with  which  other  tapes  may 
be  compared. 

To  compare  another  tape  with  the  standard  tape  the 
coefficient  of  expansion  of  the  latter  must  be  known.  To 


28.  STANDARD   TAPES.  77 

determine  this  the  tape  is  stretched  out  on  the  floor  of  a  large 
room  whose  temperature*  can  be  varied.  With  a  spring 
balance  at  each  end  it  is  Bulled  to  a  certain  tension,  the 
thermometer  noted  and  a  certain  length  marked  on  two  tin 
plates  temporarily  fastened  on  the  floor.  The  temperature 
is  then  raised  or  lowered  and  the  operation  again  repeated 
under  the  same  pull.  The  change  of  length  as  marked  on 
the  tin  plates  is  accurately  measured,  and  this  is  divided  by 
the  total  length  and  by  the  number  of  degrees  to  give  the 
coefficient  of  expansion.  The  work  should  then  be  repeated 
several  times  using  different  lengths  in  each  case,  and  the 
mean  of  the  results  be  taken  for  the  final  coefficient. 

If  a  tape  is  to  be  used  under  different  tensions  its  coeffi- 
cient of  stretch  should  also  be  determined.  The  operation 
of  doing  this  is  similar  to  that  above  described,  except  that 
the  temperature  should  be  kept  constant  and  the  pull  be 
varied.  The  change  of  length  divided  by  the  difference  of 
the  pulls  and  by  the  total  length  is  the  coefficient  of  stretch. 

Sometimes  a  tape  is  stretched  over  two  supports  A  and  B, 
and  thus  owing  to  the  sag  the  measured  distance  is  too  long. 
Let  /be  the  distance  read  on  the  tape  under  a  pull  P,  let  d 
be  deflection  or  sag  at  the  middle,  and  w  the  weight  of  the 
tape  per  linear  unit.  The 
curve  of  the  tape  is  closely  that 
of  a  parabola,  and  if  L  be  the 

n     T» 

horizontal  distance,  L  =  /  —  — j    very  nearly.      Also  taking 

moments  about  the  middle  of  the  span,  Pd  =  \wl-\l  nearly. 
Eliminating  d  from  these  two  equations  there  results 


from  which  the  true  distance  L  can  be  computed   from  the 
observed  distance  /.      If  the  distance  AB  be  subdivided  into 


78  BASE   LINES.  III. 

n  equal  parts  by  stakes  whose  tops  are  on  the  same  level  as 
those  at  A  and  B,  then 


gives  the  horizontal  distance  between  A  and  B. 

It  thus  appears  that  any  observation  of  a  distance  read  on 
a  steel  tape  may  contain  three  systematic  errors  due  to  tem- 
perature, pull,  and  sag.  Let  /  be  the  temperature  and  /  the 
pull  at  which  the  tape  is  a  standard,  let  T  be  the  temperature 
and  P  the  pull  at  which  a  measurement  /  is  taken,  let  e  be 
the  coefficient  of  expansion,  and  s  the  coefficient  of  stretch, 
let  w  be  the  weight  of  the  tape  per  linear  unit,  and  if  sag 
exists  let  n  be  the  number  of  equal  spaces  in  the  distance  /. 
Then  the  reading  /  is  to  be  corrected  by  applying  the  follow* 
ing  quantities: 

Correction  for  temperature  =  -\-  e(T  —  i)l, 
Correction  for  pull  =  -{-  s(P  —  /)/, 

Correction  for  sags  =  — 

As  an  illustration,  let  t  =  56  degrees,  /  =  16  pounds, 
^  =  0.00000703,  s  =  0.00001782,  w  =  0.0066  pounds  per 
linear  foot.  Let  a  horizontal  distance  309.845  feet  be  read 
at  a  temperature  of  49^  degrees  under  a  pull  of  20  pounds, 
there  being  7  subdivisions  in  that  distance.  Then  the  cor- 
rection for  temperature  is  —  0.0142  feet,  that  for  pull  is 
-f-  0.0221,  and  that  for  sag  is  —  0.0028  feet.  The  corrected 
measured  distance  is  then  309.850  feet. 

Lastly,  if  the  measurement  is  made  on  a  slope  it  must  be 
reduced  to  the  horizontal.  For  this  purpose  the  difference 
of  elevation  of  the  two  ends  is  found  by  leveling.  Let  //  be 
this  distance  and  L  the  length  on  the  slope,  then  the  hori- 

zontal distance   is  L\  /  I  —  -^.      For  instance  if  the  length 


29.  MEASUREMENT   WITH   A   TAPE.  79 

309.850  feet  has  2.813  feet  as  the  difference  of  level  of  the 
ends,  then  the  horizontal  distance  is  309.838  feet. 

Steel  tapes  used  in  base-ljjie  work  usually  vary  in  length 
from  300  to  500  feet.  They  have  division  marks  at  every 
50  feet,  but  near  the  ends  the  marks  are  one  foot  apart,  and 
a  finely  graduated  rule  is  used  for  reading  decimal  parts  of  a 
foot. 

Prob.  28.  A  tape  is  a  standard  at  41°  F.  when  under  16  pounds 
pull  and  no  sag,  its  coefficient  of  expansion  being  0.0000069  an^  its 
coefficient  of  stretch  0.00000195.  Find  the  pull  P  so  that  no  cor- 
rections will  be  necessary  when  measurements  are  made  at  a  tem- 
perature of  38  degrees  and  with  no  sags. 

29.  MEASUREMENT  WITH  A  TAPE. 

When  a  base  is  to  be  measured  with  precision  it  should  be 
laid  out  into  divisions,  each  shorter  than  the  length  of  the 
tape,  and  stout  posts  be  set  at  its  ends  and  at  the  points  of 
division.  In  these  posts  are  placed  metallic  plugs,  each 
having  drawn  upon  it  a  fine  line  at  right  angles  to  the  direc- 
tion of  the  base.  The  elevations  of  these  plugs  should  be 
carefully  determined  by  leveling. 

Each  division  is  then  subdivided  into  several  equal  parts 
by  light  stakes  set  in  line  and  on  grade,  the  distance  between 
the  stakes  being  fifty  feet  or  less.  The  tops  of  these  stakes 
should  be  smooth  and  rounded  so  that  friction  may  not 
prevent  the  transmission  of  a  uniform  tension  throughout  the 
tape;  on  the  top  of  each  stake  two  small  nails  may  be  driven 
to  keep  the  tape  in  position.  Instead  of  stakes  special  iron 
pins  are  sometimes  used  each  having  a  hook  to  hold  the  tape. 

The  measurement  should  be  done  on  a  cloudy  day  with 
little  wind  in  order  to  avoid  errors  due  to  change  in  tempera- 
ture. The  tape  is  suspended  over  two  plugs  and  upon  the 
intermediate  stakes  and  pulled  at  both  ends  by  spring 
balances  to  the  desired  tension.  At  one  plug  a  graduation 


80 


BASE    LINES. 


Ill 


mark  of  the  tape  is  made  to  coincide  with  the  fine  line  on  the 
plug,  and  at  the  other  end  the  distance  between  the  fine  line 
and  the  nearest  graduation  mark  is  read  by  a  closely  grad- 
uated rule.  Several  measures  of  each  division  should  be 
made  at  different  times  and  with  different  pulls  and  the  tem- 
perature be  noted  at  each  reading. 

FIELD  NOTES.     BASE  LINE  EG.     OCT.  3,  1888,  P.M. 


Divisions. 

No.  of  Sub- 
divisions. 

Diff.in  Eleva- 
tion of  Ends. 

Temperature. 

Pull. 

Measured 
Distance. 

Remarks. 

feet 

o 

Ibs. 

feet 

Ill 

7 

2.813 

51 

16 

309.865 

50.5 

18 

309.857 

50.5 

20 

309.842 

5° 

16 

309  .  8/0 

5° 

18 

309.857 

Cloudy. 

49-5 

20 

309.845 

II 

7 

5.6l8 

48 

16 

332.736 

47-5 

18 

332.727 

No  Wind. 

47-5 

20 

332.712 

47 

16 

332.740 

47 

18 

332.726 

47 

20 

332.715 

I 

6 

7.924 

47 

16 

279.850 

47 

18 

279.843 

47 

20 

279.832 

48 

16 

279.848 

48.5 

18 

279.840 

48 

20 

279.837 

The  field  notes  of  one  measurement  of  a  short  base  line 
EG,  about  922  feet  long,  will  illustrate  the  method  of  opera- 
tion. There  were  three  divisions,  designated  as  I,  II,  and 
III,  the  first  having  six  and  the  others  seven  subdivisions. 


29. 


MEASUREMENT   WITH   A   TAPE. 


81 


The  steel  tape  used  was  about  400  feet  long,  and  stated  by 
its  makers  to  be  a  standard  at  56°  Fahrenheit  when  under  a 
pull  of  1  6  pounds  ^and  having  no  sag.  Its  coefficient  of 
expansion  had  been  determined  to  be  0.00000703,  its  coeffi- 
cient of  stretch  0.00001782,  and  its  weight  per  linear  foot 
0.0066  pounds.  In  order  to  correct  the  field  results  the 
expressions  deduced  in  the  last  Article  become 


Correction  for  temperature  =  —  0.00000703(56  — 
Correction  for  pull  =  -j-  o.  00001782^  —  i6)/; 

I3 
Correction  for  sag  —  —  0.00000181  5-^; 

from  which  the  corrections  are  computed.  For  example,  in 
division  III,  where  n  =  7,  the  mean  of  the  observed  distances 
is  309.856  feet  and  this  is  taken  as  the  value  of  /  in  all  the 
corrections.  These  being  computed  the  corrected  inclined 
distances  are  found  and  their  mean  gives  309.851  feet  as  the 
inclined  length.  Lastly,  this  is  reduced  to  the  horizontal, 
and  309.838  feet  is  the  final  length  of  division  III. 

COMPUTATIONS,  DIVISION  III,  BASE  EG. 


1 

Corrections. 

Temp. 

Pull 

p 

Measured 
jDistance. 

Corrected 
Distance. 

Notes. 

Temp. 

Sag. 

Pull. 

0 

Ibs. 

feet. 

feet. 

feet. 

feet. 

feet. 

5i. 

16 

309.865 

—  .0109 

-  .0043 

O 

309  8498 

»  =  7 

50.5 

18 

309.857 

—  .OI2O 

—  -0034 

-f-  .0110 

309.8526 

50  5 

20 

309.842 

—  .0120 

—  .0028 

+  .0221 

309-8493 

h  —  2.813  ft. 

50. 

16 

309.870 

—  .0131 

-  .0043 

O 

309.8526 

50. 

18 

309.857 

—  .0131 

—  .0034 

-j-  .0110 

309.8515 

Cf=—  0.0128  ft. 

49-5 

20 

309.845 

—  .0142 

—  .0028 

-f  .0221 

309.8501 

Mean  inclined  distance       =  309  851  ft. 

Mean  horizontal  distance  =  309  838  ft. 

Proceeding  in  the  same  manner  the  corrections  were  found 
for  Divisions  I  and  II,  and  the  sum  of  the  three  mean  hori- 


82  BASE    LINES.  III. 

zontal  distances  is  922.223  feet,  which  is  the  most  probable 
length  of  the  base  line  EG  as  determined  from  the  observa- 
tions of  one  day.  Four  other  measurements  of  this  base, 
made  on  four  different  days,  gave  the  results  922.220, 
922.221,  922.226,  and  922.217  feet.  The  mean  of  the  five  is 
922.221  feet,  whose  probable  error  is  o.ooi  feet  nearly,  and 
accordingly  the  uncertainty  of  this  final  mean  is  about 
•STroViFir-  Ifc  is  thus  seen  that  work  of  a  high  degree  of  pre- 
cision can  be  done  with  a  long  steel  tape  whose  constants  are 
known. 

The  greatest  errors  in  tape-line  measurements  are  those 
due  to  errors  in  comparison  with  the  standard  and  those  due 
to  the  fact  that  ^he  temperature  of  the  metal  is  not  the  same 
as  that  of  the  air.  The  latter  error  may  be  removed  by 
making  some  measurements  when  the  temperature  is  rising 
and  others  when  it  is  falling,  and  methods  have  also  been 
devised  of  finding  the  exact  temperature  of  the  tape  by 
means  of  an  electric  current  passing  through  it;  the  former 
error  cannot  be  removed  except  by  the  use  of  different  tapes 
which  have  been  independently  compared  with  the  official 
standard. 

An  account  of  the  measurement  of  a  geodetic  base  of  3780 
meters,  or  about  2.3  miles,  by  steel  tapes  is  given  by 
Woodward  in  Transactions  of  American  Society  of  Civil 
Engineers  for  October,  1893.  It  is  concluded  that  the  prob- 
able uncertainty  in  the  final  result,  arising  from  all  sources 
except  that  of  error  in  the  tape,  cannot  exceed  ^--frorinnr- 
This  precision  was  secured  by  four  days'  work  with  twelve 
men,  most  of  the  measurements  being  made  at  night.  In 
general  it  seems  to  be  an  established  conclusion  that  precision 
in  base  measurements  may  be  secured  more  cheaply  by  the 
use  of  tapes  than  by  any  other  method. 

Prob.  29.  Correct  the  measurements  on  Division  I  of  the  above 
base  line  JEG,  and  compute  the  most  probable  value  of  its  final 
length  and  its  probable  error. 


30.  BROKEN  BASES.  83 

30.  BROKEN  BASES. 

'  -j 
A  base  line  should   be  perfectly  straight  and   its  ends  be 

intervisible,  but  cases  sometimes  arise  where  obstructions, 
like  a  river  or  swampy  land,  render  direct  measurement  im- 
practicable. In  geodetic  work  such  a  location  should  not  be 
selected  for  a  base  line,  but  in  secondary  plane  triangulation 
it  may  be  used  if  expense  is  thereby  avoided. 

The  first  case  is  where  the  base  AB  is  computed  from  two 
distances  a  and  b,  measured  along  the  lines  BC  and  AC. 
The  three  angles  of  the  triangle  are  also  measured  and 
adjusted.  The  length  of  the  base  is  then  computed  from 
=  b  cosA  -(-  a  cos^,  or  from 
=  (a  +  £)'  -  4ab  sin*$C. 
It  might  at  first  be  thought  that  A' 
the  small  angles  would  introduce  a  high  uncertainty  in  the 
computed  length,  but  on  reflection  it  is  seen  that  this  is  not 
the  case  because  two  sides  of  the  triangle  are  given,  and 
accordingly  the  uncertainty  due  to  the  angles  decreases  with 
their  sines.  For  instance,  if  a  =  b  and  if  C  =  170°,  it  will 
be  found  that  a  probable  error  of  one  minute  in  C  produces 
an  uncertainty  of  only  Tou\w  m  t^le  computed  length  of 
the  base. 

A  second  case  is  where  a  stream  crosses  the  base  line 
between  B  and  C.  Here  four 
points  are  selected  on  the  line, 
two  on  each  bank,  and  at  these 
the  angles  are  read  which  the 
base  makes  with  lines  drawn  to 
an  auxiliary  station  5.  From 
these  angles  and  the  measured 
distances  AB  and  CD  the  distance  BC  is  computed  in  two 
ways,  namely, 

-    A  °5'mA  sin^'  +  ^  sinP  sinQg.  + 

==         sinC,  sm(A  4- 


84  BASE   LINES.  III. 

and  the  angles  should  be  measured  with  such  precision  that 
these  agree  in  the  last  decimal  used  in  the  numerical  work. 

Another  method  of  procedure  in  the  last  case  is  to  measure 
only  the  angles  at  the  station  5.  Let  these  be  called  5,  ,  5,  , 
•and  5,  ,  as  shown  in  the  figure,  and  let  AB  and  CD  be  called 
a  and  b.  Then  the  distance  BC  may  be  computed  from 

pr  -        ~ 

•  Jj  Cx    — 


, 
2   COSJ  2 

where  a  —  b  is  to  be  taken  as  always  positive  and  where  y  is 
an  angle  whose  value  is  found  from 


tan>  _  _^ sin(51  +  5.)  sin(5a  +  5.) 

~  (a  -  b)"  sinS,  sinS, 

This  is  the  method  recommended  by  the  U.  S.  Coast  and 
Geodetic  Survey.  The  demonstration  of  these  formulas  may 
be  easily  made  by  applying  the  second  equation  of  Art.  23 
to  the  three  points  ABC  and  then  to  BCD,  and  equating  the 
two  expressions  each  of  which  contains  the  unknown  distance 
BC.  In  order  to  verify  the  result  another  station  S'  may  be 
selected,  the  angles  be  measured  there,  and  another  compu- 
tation be  made. 

|l|;  •)  ' 

Prob.  30.  For  the  last  case  let'jkhere  be  given  a  =  90.0242  meters, 

b  =  120.0316  meters,  S1  =  19°  41^4". 6,  S^  =  39°  20'  45". 2,  and 
•S1,,  =  26°  19'  32". 8.  Using  seven-mace  logarithms  show  that  the 
length  of  £C\$  107.8408  meters. 

31.  REDUCTION  TO  OCEAN  LEVEL. 

Geodetic  base  lines  must  be  reduced  to  mean  ocean  level 
in  order  that  perfect  agreement  may  obtain  in  the  sides  of 
triangles  computed  from  different  bases.  Let  AB  be  the 
base  whose  measured  length  is  /  and  whose  mean  elevation 
above  mean  ocean  level  is  h.  Let  ab  represent  this  ocean 
level  whose  radius  of  curvature  Ca  or  Cb  is  R.  Then, 


SI- 


REDUCTION  TO  OCEAN  LEVEL. 


from  the  two  similar  sectors,  the  value  of  ab  i 


R' 


(30 


and    therefore    the    correction    to,  be    subtracted    from    the 

adjusted  measured  length  is  Ih/R. 

For  a  long   base   this  correction 

will     be  appreciable    even    when 

the  base  is  but  a  few  feet  above 

the  mean  ocean  level. 

In  Chapter  VII  it  is  shown 
how  the  radii  of  curvature  have 
been  found  for  different  lati- 
tudes; it  is  there  seen  that  for 
common  cases  the  logarithm  of  R  may  be  found  by  taking 
the  mean  of  the  logarithms  of  R^  and  R^  given  in  Table  IV 
at  the  end  of  this  volume.  When  the  azimuth  of  the  base  is 
given  and  great  accuracy  is  required  R  should  be  computed 
from 


R 


in  which  Z  is  the  azimuth  of  the  base  line  and  Rl  and  Rt  are 
taken  from  Table  IV. 

For  example,  let  the  adjusted  measured  length  of  the  base 
be  1  8  207.3267  meters,  its  mean  height  above  ocean  level 
523.2  meters,  and  its  mean  latitude  40°  36'  '.  From  Table  IV 
the  logarithm  of  R  is  6.8044705  and  the  correction  Ih/R  is 
found  to  be  1.4943  meters,  so  that  the  length  on  the  ocean 
level  is  18  205.8324  meters.  If  the  azimuth  of  the  base  be 
75°  40',  the  more  accurate  formula  gives  the  logarithm  of  R  as 
6.8052175,  from  which  the  correction  Ih/R  is  1.4917  meters, 
so  that  the  final  length  on  ocean  level  is  18  205.8350  meters. 

As  the  lengths  of  the  base  lines  have  been  reduced  to  ocean 
level  it  follows  that  all  distances  computed  in  a  geodetic  tri- 
angulation  are  really  the  projections  of  the  actual  distances  on 


86  BASE    LINES.  III. 

the  surface  of  a  spheroid  coinciding  with  ocean  level.  Thus, 
if  /!  represent  the  computed  geodetic  distance  between  two 
stations  whose  elevation  above  ocean  level  is  //,  it  is  plain  from 
(31)  that  the  true  distance  between  those  stations  is 


For  example,  let  !1  =  100  miles  and  h  =  2\  miles ;  then, 
using  for  R  the  mean  value  3959  miles,  the  true  distance  /  is 
100.06315  miles.  Here  it  is  seen  that  the  difference  /  —  /t  is 
1/1583.6  of  the  distance  llt  so  that  the  error  in  considering  a 
computed  geodetic  distance  as  the  actual  distance  may  often 
be  a  large  one. 

Prob.  31.  A  base  line  measurement,  made  374  feet  above  ocean 
level,  gives  1725.065  feet.  What  is  the  length  of  the  base  when 
reduced  to  ocean  level? 


32.  SPIRIT   LEVELING.  8/ 


CHAPTER  IV. 
LEVELING. 

32.    SPIRIT  LEVELING. 

The  method  of  determining  differences  of  elevation  by  an 
engineer's  level  and  rod  is  called  spirit  leveling  to  distinguish 
it  from  the  method  in  which  vertical  angles  are  used.  In 
common  work  the  telescope  is  made  level  by  bringing  the 
bubble  into  the  middle  of  the  attached  scale.  In  geodetic 
work  a  sensitive  bubble  is  used  and  readings  of  its  ends  taken 
on  the  scale,  corrections  to  the  rod  readings  being  applied 
according  to  the  distance  of  the  rod  from  the  instrument. 

A  level  surface  is  one  parallel  to  that  of  a  fluid  at  rest,  and 
the  process  of  leveling  consists  in  rinding  the  elevations  of 
points  above  the  mean  surface  of  the  ocean.  The  line  of 
collimation  of  the  telescope  of  a  properly  adjusted  and  leveled 

\A' /     IB'  /C 


A  -x 

C 

instrument,  when  revolved  around  the  vertical  axis,  gen- 
erates a  plane  which  is  tangent  to  a  level  surface.  The  line 
of  sight,  however,  is  depressed  below  that  plane  owing  to 
refraction,  and  it  lies  between  the  tangent  plane  and  the  level 
surface,  but  nearer  to  the  former.  Thus  if  /  be  the  telescope 
of  the  instrument,  the  straight  line  A'B'  represents  the 
tangent  plane,  and  the  curved  line  ab  the  level  surface,  while 
the  actual  line  of  sight  is  a'b'  y  the  points  af  and  br ,  in  conse- 


88  LEVELING.  IV. 

quence  of  refraction,  appearing  to  be  in  the  tangent  plane  at 
A1  and  ff. 

The  rule  that  front  and  back  sights  should  be  of  equal 
length  in  order  to  secure  precision  is  one  that  is  well  known, 
and  the  figure  shows  the  reasons  for  it.  Let  rods  be  set  at 
A,  B,  and  C  in  order  to  find  the  heights  of  B  and  C  above 
A ;  then  the  observer  will  set  the  targets  at  a ',  br ,  and  c',  and 
the  readings  of  the  rods  will  be  Aa',  Bb' ',  and  Cc' .  The 
height  of  B  above  A  will  be  given  by  Bb'  —  Aa! ,  and  that  of 
C  above  A  will  be  given  by  Cc'  —  Aa' .  Now,  owing  to  the 
combined  effect  of  curvature  of  the  level  surface  and  of 
refraction  of  the  air,  the  errors  aa' ,  bb1 ',  and  cc'  have  been 
made  in  the  rod  readings,  but  the  difference  Bb'  —  Aa'  is  the 
same  as  Bb  —  Aa  if  the  horizontal  distances  from  B  and  A 
to  the  instrument  are  equal,  while  the  difference  Cc'  —  Aar 
is  not  the  same  as  Cc  —  Aa  if  the  rod  C  is  further  from  the 
instrument  than  the  rod  A. 

It  will  be  shown  in  Art.  37  that  the  deviation  of  a  tangent 
plane  from  a  level  surface  is  about  two-thirds  of  a  foot  at  a 
distance  of  one  mile  and  -§72*  feet  at  a  distance  of  n  miles,  also 
that  the  deviation  of  the  tangent  plane  from  the  refraction 
surface  is  one-seventh  of  that  of  the  level  surface.  The  com- 
bined effect  of  curvature  and  refraction  is  hence  to  cause  an 
elevation  of  the  line  of  sight  above  the  level  surface  amount- 
ing to  about  0.57  feet  in  one  mile  or  0.57^'  feet  in  ;/  miles; 
a  more  exact  rule  is  0.000206  feet  in  100  feet  and  0.000206^* 
feet  in  loon  feet.  Thus,  in  the  above  figure,  if  the  rods  A 
and  B  be  at  500  feet  from  the  instrument,  aa'  and  bb'  are 
each  0.0051  feet,  but  the  difference  of  level  between  A  and 
B  is  free  from  error.  If  the  rod  at  C  be  I  ooo  feet  from 
the  instrument  cc'  is  0.0206  feet,  and  hence  the  difference 
Cc'  —  Aa'  is  0.0155  feet  in  error,  for  cc*  is  0.0155  feet 
greater  than  aa' . 

Another  class  of  errors  that  is  largely  removed  by  taking 
backsights  and  foresights  of  equal  length  are  those  due  to- 


33-  DUPLICATE   LINES.  89 

0 

lack  of  perfect  adjustment  of  the  instrument.  Thus  if  the 
line  of  collimation  be  not, exactly  parallel  to  the  level  bubble 
the  reading  on  the  rod  at  An  may  be  too  great,  but  when  the 
sight  is  made  on  B  the  reading  there  is  also  too  great,  and 
hence  these  equal  errors  disappear  in  taking  the  difference  of 
the  rod  readings.  It  is  not  desirable  to  try  to  do  precise 
work  with  an  instrument  that  is  not  in  good  adjustment,  but 
it  is  essential  to  note  that  precise  work  cannot  be  done  with 
back  and  front  sights  of  unequal  length,  unless  the  lengths  of 
these  be  measured  and  a  correction  be  applied  for  the  com- 
bined effect  of  curvature  and  refraction.  In  common  work 
pacing,  or  even  estimation,  may  be  sufficient  to  prevent  the 
introduction  of  these  errors,  but  in  precise  work  the  distances 
should  always  be  measured  to  the  nearest  foot. 

Prob.  32.  Let  the  rod  readings  at  A,  B,  and  C  be  1.073,  3-I37> 
and  9.271  feet,  the  distances  from  the  instrument  being  200,  250, 
and  400  feet.  Find  the  elevations  of  B  and  C  above  A. 

33.  DUPLICATE  LINES. 

In  common  work  with  an  engineer's  level  the  precision  of 
the  elevations  of  the  bench  marks  may  be  increased  by 
running  a  second  line  between  them  and  then  taking  the 
mean  of  the  differences  of  level.  This  precaution  can  never 
be  neglected  in  good  work,  for  one  measurement  affords  no 
data  for  estimating  the  precision  of  the  results,  It  is  better 
to  run  the  two  lines  in  opposite  directions  rather  than  in  the 
same  direction. 

Semi-duplicate  lines  are  those  run  in  the  same  direction, 
having  the  same  bench  marks  and  heights  of  instruments  but 
different  turning  points.  Two  sets  of  notes  are  kept  which 
are  not  compared  until  a  check  is  made  on  a  bench  mark. 
Thus  in  the  figure  let  M  and  N  be  two  bench  marks  and 
/,  ,  7a ,  /, ,  and  74  the  points  where  the  level  is  set  up,  while 
AU  At,  and  A3  are  the  turning  points  on  line  A,  and  Bl , 


90  LEVELING.  IV. 

BV  and  7?,  are  the  turning  points  on  line  B.  The  instrument 
being  set  at  II  a  backsight  is  taken  on  M  and  recorded  in  the 

^ __A__ 

M     ^:==ir::^  /4 

notes  for  line  A  ;  then  another  backsight  is  taken  on  M  and 
recorded  in  the  notes  for  line  B.  The  two  turning  points  Al 
and  Bl  having  been  selected,  foresights  are  taken  upon  them 
in  succession  and  the  readings  recorded  in  the  notes  for  lines 
A  and  B  respectively.  Then  the  instrument  is  moved  to  73 
and  backsights  taken  on  A,  and  Bl  which  are  recorded  in  the 
separate  notes  for  A  and  B.  On  arriving  at  74  backsights  are 
taken  upon  A3  and  Bt  and  two  foresights  upon  N.  Thus  two 
lines  MA^A^A^N  and  MB,B^BtN  have  been  run  between  the 
bench  .marks  M  and  N;  if  the  elevation  of  TV  is  to  be  deter- 
mined from  the  given  elevation  of  M,  two  sets  of  observa- 
tions are  at  hand  from  whose  comparison  and  combination  it 
can  be  obtained  with  a  higher  degree  of  precision  than  by  a 
single  line. 

Another  method  of  running  semi-duplicate  lines  is  to  have 
the  same  turning  points  but  different  heights  of  instruments. 
Thus,  in  the  above  figure,  if  7,  be  a  bench  mark  the  level  is 
set  at  Al ,  a  backsight  taken  upon  II  and  a  frontsight  upon 
7,;  then  the  instrument  is  set  at  B^  ,  a  backsight  taken  upon 
7,  and  a  frontsight  upon  7, .  This  method  is  not  as  conven- 
ient or  expeditious  as  that  above  described,  since  it  involves 
two  rodmen,  and  it  would  be  better  to  run  two  independent 
duplicate  lines  in  opposite  directions  between  the  bench 
marks. 

By  taking  proper  precautions  to  preserve  equality  in  the 
lengths  of  back  and  front  sights,  shading  the  instrument  from 
the  rays  of  the  sun,  and  keeping  the  rod  truly  vertical,  semi- 
duplicate  lines  may  be  run  with  an  engineer's  level  so  that 
the  probable  error  of  differences  in  elevation  shall  be  less  than 
0.005  feet  for  bench  marks  one  mile  apart.  In  precise  level- 


34-  PROBABLE   ERRORS   AND   WEIGHTS.  Ql 

ing  where  readings  are  taken  to  ten-thousandths  of  a  foot, 
the  probable  error  may  be,  made  much  smaller.  The  adjusted 
elevations  of  the  benches  arfi  of  course  the  mean  of  the  values 
found  by  the  two  lines. 

It  is  sometimes  observed  that  the  elevations  found  by  one 
line  tend  to  be  greater  than  those  found  by  the  other.  For 
example,  a  line  of  semi-duplicate  levels  run  from  Bethlehem 
to  Allentown,  Pa.,  by  students  of  Lehigh  University  in  1894 
may  be  briefly  noted.  The  total  distance  was  32  750  feet, 
this  being  divided  into  27  sections  with  28  bench  marks. 
Computing  the  27  differences  of  level  for  lines  A  and  B  it  was 
found  that  nine  were  the  same  for  both,  that  line  A  had  nine 
greater  and  also  nine  less  than  line  B\  computing  the  eleva- 
tions of  the  27  benches  from  that  of  the  Bethlehem  bench  it 
was  found  that  25  of  these  were  greater  on  line  B  than  on 
line  A.  The  discrepancy  between  the  two  lines  reached  a 
maximum  of  0.009  ^eet  at  18000  feet  from  the  Bethlehem 
bench,  then  decreased  to  o.ooi  feet,  and  afterwards  increased 
until  it  became  0.005  ^eet  at  ^e  Allentown  bench.  The 
probable  error  of  the  difference  of  level  between  the  end 
benches,  computed  by  the  method  of  the  next  Article,  was 
found  to  be  0.004  feet-  This  is  perhaps  a  little  smaller  than 
would  be  found  by  independent  duplicate  lines  run  in  opposite 
directions. 

Prob.  33.  The  difference  of  level  of  two  points  Pl  and  /*,  was 
found,  by  setting  the  level  half-way  between  them,  to  be  6.438  feet. 
A  second  observation  gave  6.436  feet,  and  a  third  one  gave  6.437 
feet.  Show  that  the  probable  error  of  a  single  observation  was 
0.0007  feet. 

34.  PROBABLE  ERRORS  AND  WEIGHTS. 

The  probable  error  of  the  difference  in  elevation  of  two 
bench  marks  increases  with  the  number  of  times  the  instru- 
ment is  set  up  between  them,  and  will  hence  be  greater  in  a 
hilly  region  than  in  a  prairie  country.  It  will  also  depend 


92  LEVELING.  IV. 

upon  the  precision  of  the  instrument  and  upon  the  skill  of 
the  leveler  and  rodman,  so  that  different  classes  of  work  will 
have  different  probable  errors. 

Assuming  that  the  instrument  is  set  up  about  the  same 
number  of  times  in  a  distance  of  one  mile  or  one  kilometer, 
it  will  be  clear  that  the  probable  error  in  leveling  is  governed 
by  the  same  law  as  that  for  linear  measurements,  namely  that 
it  increases  as  the  square  root  of  the  distance.  Thus  if  rl  is 
the  probable  error  in  leveling  a  distance  of  unity,  say  one 
mile  or  one  kilometer,  then  the  probable  error  in  leveling  the 
distance  /  is  r  =  rl  V~l.  Thus  if  the  probable  error  for  a  line 
one  mile  long  is  0.006  feet  the  probable  error  for  a  line  four 
miles  long  is  0.012  feet. 

By  means  of  duplicate  lines  of  levels  the  probable  error  r, 
may  be  obtained  by  the  application  of  formula  (26),  the 
weights  being  taken  as  the  reciprocals  of  the  lengths  of  the 
lines.  Semi-duplicate  lines,  like  those  described  in  the  last 
Article,  may  be  used  for  the  same  purpose,  but  probably  the 
value  of  r,  found  from  them  is  somewhat  smaller  than  from 
two  lines  run  in  opposite  directions.  As  an  example  of  the 
method,  let  Da  and  Db  be  the  differences  of  elevation  be- 
tween two  bench  marks  as  determined  by  the  two  lines, 
d  the  differences,  or  discrepancies,  between  these,  /  the  dis- 
tance between  the  benches,  and  p  the  weight  of  d  in  terms 
of  the  weight  of  the  unit  of  distance.  *  Taking  the  following 
five  measurements,  and  regarding  1000  feet  as  the  unit  of 
distance,  the  sum  of  the  five  values  of  pd*  is  0.0000404 


Da 

—  3.801    — 

13.429   — 

0.363   + 

5.528 

+  9-657  feet 

Db 

-  3-803 

13.426 

0.365   + 

5.532 

-f  9.653  feet 

d 

-f  O.OO2 

0.003    + 

0.002     — 

O.OO4 

+  0.004  feet 

I 

0.400 

0.840 

I.5OO 

1.800 

2.000  feet/  1  ooo 

p 

2.50 

1.19 

0.67 

0.56 

0.50 

pd* 

0.0000  1  00 

0.0000107 

0.0000027 

0.00000 

90  0.000000080 

and  then   from   formula  (26)  the   value  of  rl  is  found  to   be 
0.0014  feet-      Thus,  for  this  class  of  work,  the  probable  error 


34- 


PROBABLE    ERRORS   AND   WEIGHTS. 


93 


in  leveling  a  distance  of  I  ooo  feet  is  0.0014  ^eet»  and  hence 
the  probable  error  in  leveling' any  distance  is  6.0014  1/7,  where 
/is  the  distance  in  thousand  of  feet.  To  find  the  probable 
error  for  one  mile  /Is  to  be  taken  as  5.28,  and  thus  0.0032  Vn 
expresses  the  probable  error  of  a  line  of  levels  n  miles  in 
length. 

As  weights  are  inversely  proportional  to  the  squares  of 
probable  errors  it  follows  that  the  weights  of  differences  of 
elevation  are  inversely  proportional  to  the  distances  over 
which  the  leveling  is  extended. 
For  example,  let  there  be  run 
three  routes  from  P  to  Q  giving 
the  results 

Route.     Miles.       P  above  Q. 

1  5         37-407  feet 

2  6         37-392  feet 

3  10         37.4H  feet 

If  the  precision  of  the  work  per  mile  is  the  same,  the  value 
of  rl  being  the  same  for  the  three  lines,  then  the  weights  of 
the  three  results  are  to  be  taken  as  ^,  ^-,  and  -^.  The 
adjusted  elevation  of  P  above  Q  is  then  found  by  the  rule 
of  Art.  4  to  be  37.403  feet. 

The  probable  error  rl  may  be  also  computed  from  lines  run 
between  two  benches  by  different  routes,  as  in  the  last 
example.  The  method  to  be  followed  is  that  of  formula 
(9)" '.  Thus,  taking  the  weight  of  a  line  one  mile  long  as 
unity,  the  residuals  v  are  found  and  the  sum  2pv*  is 


V  V 

—  0.004  0.000016 

-f-O.OII  O.OOOI2I 

—  O.OI  I  O.OOOI  2  I 


0.20 


0.10 


0.00000320 
0.00002057 

O.OOOOI2IO 


z  =  37.403  0.000258  0.00003587  = 

formed.      Then,  from  the  formula,  rl  =  0.0029  ^eet»  which  is 
the  probable  error  of  a  difference  of  level  found  from  a  line 


CIVIL  ENGINEERING 


94  LEVELING  IV. 

one  mile  long.  Finally,  the  probable  errors  of  the  three 
observed  differences  of  level  are  found  from  the  square-root 
rule  to  be  0.0065,  0.0070,  and  0.0092  feet,  while  the  probable 
error  of  the  adjusted  elevation  is  0.0039  feet,  so  that  37.403 
±  0.004  feet  may  be  written  as  the  final  result. 

Prob.  34.  If  the  probable  error  in  leveling  one  mile  is  0.003  feet, 
what  is  the  probable  error  in  a  line  one  kilometer  long,  and  also  in 
a  line  100  kilometers  long? 

35.  ADJUSTMENT  OF  A  LEVEL  NET. 

When  a  closed  circuit  is  made  by  running  from  A  around 
to  A,  leaving  the  benches^,  C,  and  D,  the  adjusted  eleva- 

tions  of  these  are  to  be  made  by 
distributing  the  error  of  closure  in 
direct  proportion  to  the  distances 
)  between  the  benches.  For  example, 
starting  from  A  with  the  correct 
elevation  of  420.317  feet  above  mean 
ocean  level,  the  following  elevations  of  other  benches  are 
found,  and  on  returning  to  A  its  elevation  is  420.467  feet, 
showing  a  discrepancy  of  o.  150  feet.  The  distances  between 
the  benches  being  6,  3,  4,  and  2  miles,  T6T  of  the  discrepancy 
is  to  be  subtracted  from  the  elevation  of  B,  T9y  from  that  of 
C,  and  so  on.  This  method  of  adjustment  is  one  that  would 

Correction. 

o.ooo 

—  0.060 

—  0.090 

—  o. 130 

—  o. 150 

be  naturally  used  by  every  one,  and  it  will  be  seen  that  it 
agrees  with  the  results  obtained  by  the  application  of  the  rule 
in  Art.  21  to  the  determination  of  the  most  probable  differ- 
ences of  the  elevations  between  the  benches. 


Bench. 

Miles 
from  A. 

Observed 
Elevation. 

Adjusted 
Elevation. 

A 

0 

420.317 

420.317 

B 

6 

532.918 

532.858 

C 

9 

607.200 

607  .  i  10 

D 

'3 

5!°-3T5 

510.  185 

A 

15 

420.467 

420.317 

35.  ADJUSTMENT   OF  A   LEVEL   NET.  95 

A  net  of  levels  consists  of  several  lines  connecting  benches 
in  such  a  manner  that  the,  elevation  of  one  can  be  deduced 
from  another  by  several  different  routes.  An  example  of  the 
method  of  adjustment  is  given  in  Arts.  5  and  6,  where,  how- 
ever, the  weights  of  the  different  results  are  taken  as  equal. 
By  introducing  the  weights  according  to  the  method  of 
Art.  7,  taking  them  as  inversely  proportional  to  the  lengths 
of  the  lines,  the  same  process  may  be  applied  to  any  given 
case.  For  example,  take  the 
case  shown  in  the  figure  where 
eight  differences  of  elevation 
between  six  benches  are  ob- 
served in  a  net  consisting  of 
three  closed  figures.  These 
three  figures  give  three  geomet- 
ric conditions  and  accordingly 
there  can  be  but  five  independ- 
ent quantities  in  the  observation 
equations.  This  is  perhaps  seen  more  clearly  by  noting  that, 
if  the  elevation  of  one  bench  be  given,  the  elevations  of  the 
five  others  are  to  be  obtained.  In  general  the  number  of 
independent  quantities  in  any  net  of  level  lines  is  one  less 
than  the  number  of  benches. 

For  example,  let  the  eight  observed  differences  of  elevation 
be  as  given  below,  their  weights  being  taken  as  the  reciprocals 
of  the  distances  between  the  benches.  Let  v.  ,  v~ ,  v. ,  v* , 


No. 

Benches. 

Observed  h. 
Feet. 

Distance. 
Miles. 

Weight. 

Adjusted  h. 
Feet. 

I 

B  above  A 

12  .02 

4.0 

0.25 

12.039 

2 

C  above  B 

23.06 

7.2 

o.  14 

23.012 

3 

D  above  C 

14.30 

5.0 

0.20 

14.340 

4 

D  above  F 

29.44 

6-3 

o.  16 

29.389 

5 

C  above  F 

15  .02 

2.0 

0.50 

15.049 

6 

F  above  E 

9-34 

4.8 

O.2I 

9-372 

7 

B  above  E 

'•45 

3-5 

0.29 

1  .410 

8 

E  above  A 

10  67 

8-3 

0.12 

10.630 

96  LEVELING.  IV. 

and  v,  be  corrections  to  be  applied  to  the  observed  values 
^i  >  h, 1  ^3  >  hb  >  and  //7  in  order  to  give  the  most  probable 
values.  Then  the  eight  observations  are  expressed  in  terms 
of  these  five  quantities,  the  condition  //,  —  /z4  -f-  hb  =  o  giving 
the  fourth  observation  in  terms  of  z>,  and  t/4;  thus, 

1.  +  v,  =        o, 

2.  +  7.',  =  O, 

3.  +  ^8  =        o, 

4-  +   Vt  +   V.  =   +  0.12, 

5.  +  v.  o, 

6.  +  v,  -  vb  -  v,  =  -  0.15, 

7.  +  v,  =        o, 

8.  +  ^i  —  ^T  =  +  o.  10. 

From  these  the  normal  equations  are  formed  by  the  rule  of 
Art.  7,  using  the  given  weights,  and  their  solution  furnishes 
the  values  v[  =  +0.019,  vt  =  —  0.048,  etc.,  from  which  the 
above  adjusted  most  probable  values  of  the  five  quantities  are 
found.  Then  the  values  of  7z4 ,  ht ,  and  h6  immediately  result. 

The  probable  error  for  a  line  of  levels  one  mile  long  can 
now  be  computed  by  formula  (10).  Each  of  the  corrections 
being  squared  and  multiplied  by  its  weight,  2pv*  is  found  to 
be  0.00246;  then  as  n  =  8  and  g  =  $,  there  results  rl  = 
0.019  feet  as  tne  probable  error  in  leveling  one  mile,  and 
accordingly  0.019  ^  ls  tne  probable  error  in  leveling  /  miles. 
The  degree  of  precision  of  the  levels  in  this  net  is  hence 
quite  low  compared  with  that  required  for  city  work. 

Another  method  of  stating  observation  equations  in  the 
above  case  is  to  take  the  elevations  of  five  benches  as  the 
quantities  to  be  found.  Thus,  if  the  elevation  of  A  be 
given,  approximate  values  of  the  elevations  of  the  others  are 
readily  found,  and  the  corrections  to  be  applied  to  these  may 
be  called  vbt  ve1  etc.  Then  each  observation  is  expressed  in 
terms  of  these  corrections,  and  their  most  probable  values  are 
found  by  the  solution  of  the  resulting  normal  equations. 


36.  GEODETIC    SPIRIT   LEVELING.  9? 

The  adjusted  elevations  will  be  the  same  as  those  derived 
from  the  adjusted  differences  that  are  given  above.  Thus  if 
the  elevation  of  A  fee  312.724  feet,  that  of  B  will  be  324.763 
feet,  that  of  E  will  be  323.353  feet,  and  so  on. 

Prob.  35.  The  elevation  of  a  bench  P  is  725.038  feet.  Level 
lines  run  between  it  and  the  benches  Q,  -tf,  and  S,  give  the  following 
observations: 

No.  Benches.  Difference  in  Elevation.         Distance. 

Feet.  Miles. 

1  B  above  A  35.080  3 

2  C  below  A  8.698  6 

3  D  below  A  19-9°S  4 

4  C  above  D  11.212  3 

5  C  below  B  43-78°  3 

6  B  above  D  54  "995  6 

State  the  observation  equations,  form  and  solve  the  normal  equa- 
tions, and  find  the  adjusted  elevations  of  the  benches. 

36.   GEODETIC  SPIRIT  LEVELING. 

Engineers'  levels  are  of  two  types,  the  Y  level  and  the 
dumpy  level,  the  former  being  easier  of  adjustment  while  the 
latter  is  more  precise.  A  dumpy  level  with  two  vertical  and 
three  horizontal  wires  in  the  diaphragm  of  its  telescope,  and 
having  also  a  sensitive  bubble,  may  be  called  a  geodetic  instru- 
ment. The  rod  is  to  be  brought  into  the  field  between  the 
two  vertical  wires,  and  readings  taken  upon  it  by  each  of  the 
horizontal  wires  or  by  the  help  of  a  micrometer  screw.  The 
limits  of  the  ends  of  the  bubble  are  read  upon  the  attached 
scale.  The  rod  is  provided  with  attached  levels  for  securing 
verticality  and  it  is  set  on  a  foot-plate  planted  in  the  ground. 
The  distances  from  the  instrument  to  the  backsight  and  front- 
sight  positions  of  the  rod  are  measured. 

Each  instrument  must  be  tested  at  intervals  in  order  to 
determine  the  angular  distance  between  the  wires  and  the 
angular  value  of  one  division  of  the  bubble  scale.  The  usual 


98  LEVELING.  IV. 

adjustments  for  the  level  bubble  and  collimation  axis  are  to 
be  made,  as  also  a  series  of  measurements  for  determining 
the  small  errors  still  remaining  in  them.  With  these  data 
tables  can  be  made  out  for  reducing  the  readings  of  each  wire 
to  the  middle  wire,  for  eliminating  the  error  of  inclination  as 
determined  by  the  readings  of  the  ends  of  the  bubble,  and 
for  eliminating  the  error  of  collimation. 

As  precise  leveling  for  geodetic  surveys  is  generally  done 
in  the  metric  system,  the  constants  of  Art.  32  are  not  directly 
applicable  for  the  elimination  of  errors  caused  by  unequal 
lengths  of  back  and  fore  sights.  If  these  distances  be  in 
meters  and  /,  and  /,  their  values,  the  former  being  the  greater, 
then,  for  usual  atmospheric  conditions, 

d  =  0.0000675  (A'  -  '.')  (36) 

is  the  correction  in  millimeters  to  be  subtracted  from  the 
difference  in  elevation  hl  —  h^.  For  instance,  if  /,  =  200 
meters  and  /„  =  170  meters,  then  the  difference  hl  —  //„  as 
found  from  the  rod  readings  is  0.75  millimeters,  or  0.00075 
meters,  too  large.  If  the  distances  be  in  feet,  then 

d  =  0.000206  (/,'  —  /,')  (36)' 

is  the  correction  in  thousandths  of  a  foot  to  be  subtracted 
from  the  difference  hv  —  //,;  thus  if  /,  =  656  feet  and  /„  = 
558  feet,  d  is  2.5  thousandths  of  a  foot  or  0.0025  fee*.  These 
formulas  are  demonstrated  in  Art.  37. 

The  running  of  a  line  of  geodetic  levels  is  necessarily  slow 
work,  for  daily  tests  of  the  instrument  must  be  made  and 
corrections  applied  to  every  rod  reading  in  order  to  remove 
the  errors  above  mentioned.  The  line  is  divided  into  sections 
from  five  to  ten  miles  in  length,  each  of  which  is  leveled  in 
opposite  directions.  The  probable  error  of  the  elevations  of 
the  bench  marks  found  by  combining  the  two  sets  of  observa- 
tions has  been  made  less  than  two  millimeters  for  a  distance 
of  one  kilometer,  which  is  equivalent  to  about  0.008  feet  for 
a  distance  of  one  mile. 


37.  REFRACTION   AND   CURVATURE.  99 

Notwithstanding  the  apparent  accuracy  of  leveling  by  one 
of  the  instruments  above  described,  the  item  of  cost  is  so 
high  that  it  cannot  be  usecf  except  on  government  work.  It 
may  be  remarked,  further,  that  the  probable  errors  deduced 
from  the  discussion  of  such  level  lines  are  but  little,  if  any, 
less  than  those  that  can  be  obtained  by  good  work  under  the 
common  method.  By  rerunning  the  sections  several  times 
by  the  semi-duplicate  plan  of  Art.  34,  using  a  good  engineer's 
dumpy  level,  and  eliminating  the  systematic  errors  by  making 
equal  the  lengths  of  backsights  and  foresights,  it  is  not  diffi- 
cult to  secure  results  whose  probable  errors  shall  be  as  low 
or  lower  than  those  of  the  so-called  geodetic  method,  while 
the  cost  of  the  work  per  mile  will  be  less  than  half  as  great. 

Prob.  36.  Consult  Wilson's  paper  on  "  Spirit  Leveling  "  in  Trans- 
actions of  American  Society  of  Civil  Engineers,  1898,  Vol.  xxxix, 
and  Molitor's  paper  in  Vol.  XLV,  1901.  Compare  methods  used  by 
the  different  U.  S.  Government  Surveys,  and  collect  facts  regarding 
the  cost  of  running  long  lines  of  levels. 

37.  REFRACTION  AND  CURVATURE. 

When  light  travels  through  air  of  varying  density,  its  path 
is  a  curved  line.  If  the  surface  of  the  earth  were  a  plane  a 
ray  of  light  moving  horizontally  would  suffer  no  refraction 
since  the  air  would  be  of  uniform  density  at  all  points  in  its 
path.  Owing  to  the  curvature  of  the  earth  a  ray  of  light 
passing  from  c'  to  7,  in  the  figure  of  Art.  32,  travels  through 
air  of  increasing-  density  because  c  is  further  than  /  from  the 
level  surface;  similarly  light  passing  from  c  to  /tends  to  do 
so  in  a  straight  line,  but  encountering  denser  air  its  path 
becomes  a  curve  which  lies  between  the  chord  cl  and  the  arc 
cl.  Hence  refraction  is  a  consequence  of  curvature. 

To  develop  formulas  for  the  effect  of  curvature  and  refrac- 
tion, it  is  necessary  to  take  for  granted  that  the  earth  is  9 
globe  whose  mean  radius  R  is  about  3  959  miles  or  6371  kilo' 


100  LEVELING.  IV. 

meters.  Let  AO  and  BO  be  this  radius  in  the  exaggera- 
ted figure,  AC  a  short  distance  /  which  is  sensibly  equal 
to  the  tangent  ABy  and  bA  the  path  in  which  light  travels 

from  b  to  A,  thus  making  b  appear 
at  B  to  an  observer  at  A.  The  de- 
viation due  to  curvature  in  the  dis- 
tance /  is  hence  represented  by  BC 
and  that  of  refraction  by  Bb,  their 
difference  bC  being  the  combined  de- 
viation ;  let  these  be  called  c,  kc,  and 
d  respectively,  k  being  an  abstract  number  less  than  unity 
whose  value  will  be  shown  later  to  be  about  ^.  Thus  d 
is  expressed  by  (i  —  k)c. 

The  value  of  c  is  readily  found,  from  the  right-angled  tri- 
angle ABO,  to  be  given  by  2Rc  +  S  =  /",  or  since  c  is  very 
small  its  square  may  be  neglected,  and  thus 

r 

C=TR 

is  the  deviation  of  the  tangent  plane  from  the  level  surface. 
The  combined  deviation  due  to  curvature  and  refraction,  of 
the  distance  bC,  is  then 

*=(l-*>5-  (37) 

From  this,  using  for  k  the  mean  value  ^,  there  is  found 
d=  0.00000006757',     (d  and  /  in  meters), 


ers),  )       .     ., 
,       )  ' 


d  =  0.00000002  o6/a,     (d  and  /  in  feet), 
from  which  the  formulas  in  Art.  38  directly  result. 

If  the  elevation  of  the  eye  of  an  observer  above  the  ocean 
is  known  the  distance  to  the  sea  horizon  may  be  deduced 
from  (38).  Thus,  for  different  systems  of  measures, 


/(in  kilometers)        =  3.85  Vd(\n  meters), 
/(in  statute  miles)    =1.32  Vd  (in  feet), 
/  (in  nautical  miles)  =1.13  \' d  (in  feet). 


38.  VERTICAL  ANGLES.  IOI 

These  results,  like  all  in  .this  Article,  are  mean  ones,  as 
curvature  varies  in  different  latitudes,  and  refraction  varies 
under  different  atmospheric^'conditions. 

Prob.  37.  If  the  elevation  d  above  the  sea  horizon  is  given    in 
meters,  what  is  the  formula  for  /  in  nautical  miles  ? 


38.  VERTICAL  ANGLES. 

The  effect  of  refraction  on  any  vertical  angle  is  to  render 
the  measured  value  too  large  or  too  small  according  as  it  is 
an  angle  of  elevation  or  angle 
of  depression,  while  curvature 
produces  the  opposite  effect. 
In  the  figure  let  A  and  B  be 
two  stations  whose  horizontal 
distance  apart  is  /,  the  station 
B  being  higher  than  A.  In 
order  to  find  the  difference  in  elevation  the  vertical  angle  of 
elevation  BAG,  or  the  vertical  angle  of  depression  ABD,  is 
needed.  Let  an  instrument  be  set  at  A  and  its  horizontal 
limb  be  made  tangent  to  the  level  surface  ARC  in  the  direc- 
tion Ae\  in  consequence  of  refraction  the  station  B  appears  to 
be  in  the  direction  Af,  and  fAe  is  the  measured  angle  of 
elevation.  The  measured  value  is  thus  too  large  by  the 
refraction  angle  fAB  and  too  small  by  the  curvature  angle 
eAC\  the  true  required  angle  BAC  is  hence/>^  —  fAB  -f- 
eAC.  In  the  same  manner,  when  the  instrument  is  set  at  By 
the  measured  angle  of  depression  isf'Be',  which  is  too  small 
by  the  refraction  angle  ABf  and  too  large  by  the  curvature 
angle  DBe' .  These  effects  of  refraction  and  curvature  are 
small,  and  sensibly  the  same  at  A  and  B  under  similar  atmos- 
pheric conditions.  Thus  the  combined  effect  of  refraction 
and  curvature  renders  the  measured  angle  at  A  too  small  and 
that  at  B  too  large  by  the  same  number  of  seconds. 

Let  a  be  the  angle  of  elevation  at  A  and  ft  the  angle  of 


102  LEVELING.  IV. 

depression  at  B,  and  let  d'r  be  the  correction  in  seconds,  so 
that  BAG  —  ex  +  d"  and  ADD  =  ft  —  d"  are  the  true 
required  angles.  In  the  last  Article  the  linear  correction  d 
normal  to  /was  found.  The  corresponding  angle  in  radians 
is  d/l  and  the  corresponding  value  in  seconds  is  206  265^/7, 
and  accordingly 

</"  =  206  265(1  -  k)^  (38) 

is  the  correction  to  be  added  to  a  and  subtracted  from  ft. 
Using  the  mean  value  k  =  1,  and  a  mean  value  of  R,  there 
results 

d"  =  O.OI394/,      when  /is  in  meters, 

d"  =  0.0042 5/,  when  /is  in  feet, 
as  the  number  of  seconds  to  be  added  to  angles  of  elevation 
and  subtracted  from  angles  of  depression.  Thus  if  the  angle 
of  elevation  of  a  station  15000  feet  distant  be  observed 
to  be  2°  19'  07"  the  true  required  vertical  angle  is  about 

2°20/   If". 

It  is  now  to  be  shown  how  the  coefficient  of  refraction  k 
can  be  found.  As  the  angles  a  -\-  d'1  and  ft  —  d"  are  equal 
the  value  of  d" ,  if  a  and  ft  are  simultaneously  measured,  is 
d"  =  £(ft  —  a).  Equating  this  to  the  former  general  value 
there  is  found  for  the  coefficient  of  refraction 


206265  /' 

in  which  ft  —  a  must  be  expressed  in  seconds.  For  instance 
let  a  =  2°  24'  58". 9,  ft  =  2°  35'  34//.2,  and  /=  23661 
meters;  then  using  for  R  the  mean  value  6371  kilometers 
there  is  found  k  —  o.  165.  A  better  value  of  R  is  that  of  the 
radius  of  curvature  of  the  level  surface  through  the  lower 
station  A.  Numerous  simultaneous  observations  of  vertical 
angles  of  elevation  and  depression  have  established  that  k 
varies  from  0. 12  to  0.18,  a  mean  value  frequently  used  being 
k  =  0.143  =  \.  The  average  values  deduced  by  the  U.  S. 
Coast  and  Geodetic  Survev  are  k  •=.  0.158  across  parts  of  the 


38.  VERTICAL  ANGLES.  1 03 

sea  near  the  coast,  and  k  =.  o.  130  between  primary  triangula- 
tion  stations  at  a  high  elevation. 

If  the  elevation  6f  the  eye  of  an  observer  above  the  ocean 
is  known  the  dip  of  the  sea  horizon  in  seconds  may  be 
expressed  by  combining  the  above  value  d"  with  that  of  / 
given  at  the  end  of  the  last  Article.  Thus  results  (38)" 


d1'  (in  seconds)  =  58.8  Vd(m  feet)  =  106.5  ^d  (in  meters). 

Also  if  the  angle  of  depression  of  the  sea  horizon  be  measured, 
ks  distance  from  the  eye  may  be  obtained  from  (38)"  and  will 
be  found  to  be  4.30  kilometers,  2.67  statute  miles,  or  2.32 
nautical  miles  for  each  minute  of  vertical  angle.  These 
results  are  mean  rough  ones,  since  both  curvature  and  refrac- 
tion vary  in  different  latitudes. 

Vertical  angles  for  determining  heights  are  usually  small, 
and  hence  a  large  probable  error  may  occur  in  a  computed 
height,  even  when  the  probable  error  of  the  vertical  angle  is 
not  large.  The  formula  h  —  /  tan  a  gives  the  height  h  in 
terms  of  the  observed  quantities  /  and  a.  Let  /  be  supposed 
to  be  without  error  and  let  r  be  the  probable  error  in  a,  then 
the  probable  error  in  h  is  rdk/da,  or  /r/cosV,  which  is  prac- 
tically Ir,  since  cos  a  is  nearly  unity.  If  r  be  expressed  in 
seconds  the  corresponding  probable  error  in  //  is  Ir/2o6  265, 
or  if  r  be  expressed  in  minutes  the  corresponding  probable 
error  in  h  is  /r/3  438.  Thus,  if  the  probable  error  of  a  vertical 
angle  be  one  minute,  and  the  horizontal  distance  //  be  6  876 
feet,  the  probable  error  in  the  computed  height  h  is  2  feet. 
In  geodetic  work,  where  leveling  by  this  method  is  done 
between  stations  many  miles  apart,  it  is  seen  that  the  prob- 
able errors  in  the  vertical  angles  must  be  rendered  very  low 
in  order  that  the  computed  heights  may  have  a  fair  degree 
of  precision.  The  uncertainty  in  a  computed  height  is 
Ir/2o6  265/1,  if  r  be  in  seconds;  for  example,  if  /=  10352 
feet  and  a  =  3°  oo'  53"  ±  03",  then  h  =  545.19  ±0.15  feet, 
and  the  uncertainty  of  h  is  about 


104  LEVELING.  IV. 

Prob.  38.  If  the  probable  error  of  /  be  rl  and  that  of  a  be  rt  show 
that  the  square  of  the  probable  error  of  h  is  found  by  r*  tan2  a  -f- 
r,Va,  where  the  last  term  must  be  divided  by  206  265*  if  ra  is  in 
seconds. 


39.  LEVELING  BY  VERTICAL  ANGLES. 

Leveling  with  the  stadia  and  transit  is  often  done  in 
topographic  work,  and  with  care  will  give  results  whose 
probable  error  should  be  not  greater  than  0.5  feet  in  one  mile 
or  than  0.5  Vn  feet  in  n  miles.  A  greater  degree  of  precision 
can  be  secured  by  measuring  the  horizontal  distances  with  a 
tape,  reading  the  vertical  angles  to  half-minutes,  selecting 
the  stations  so  that  the  angles  of  depression  arc  about  equal 
in  number  to  the  angles  of  elevation,  and  having  a  fair 
uniformity  in  the  lengths  of  backsights  and  foresights.  In  no 
case,  however,  can  this  work  attain  a  degree  of  precision 
comparable  with  that  done  by  spirit  leveling. 

The  difference  in  elevation  of  two  stations  of  a  triangula- 
tion  can  be  computed  when  the  horizontal  distance  between 
them  has  been  obtained.  The  best  method  is  to  make 
simultaneous  observations  of  the  angles  of  elevation  and 
depression.  Using  the  notation  of  the  last  Article,  it  is  seen 
that  a.  -f-  d"  and  fi  —  d"  are  the  true  angles  required,  or 
since  these  values  are  equal  the  true  vertical  angle  is  expressed 
by  \(a  -J-  /?),  and  hence 

A  =  t  tan£(a  +  ft) 

is  the  required  difference  in  elevation.  It  is  thus  seen  that 
the  effects  of  curvature  and  refraction  are  eliminated  by 
taking  the  mean  of  the  two  observed  vertical  angles  a  and  /?. 
In  this  method  it  is,  however,  essential  that  the  two  measure- 
ments should  be  made  as  nearly  simultaneously  as  possible  in 
order  that  the  same  atmospheric  conditions  may  affect  both 
angles,  for  it  is  found  that  the  coefficient  of  refraction  varies 
•with  temperature  and  barometric  pressure. 


39-  LEVELING    BY   VERTICAL  ANGLES.  IO5 

In  a  geodetic  trkngulation  measurements  of  vertical  angles 
are  carried  on  at  the  sam,e  "time  with  those  of  the  horizontal 
angles,  and  it  is  not  usually  possible  that  the  vertical  angles 
at  two  stations  can  be  simultaneously  measured.  Records  of 
the  weather  are  kept,  however,  and  by  taking  at  each  station 
a  considerable  number  of  observations  it  is  possible  to  select 
for  any  two  stations  several  which  are  made  under  like 
atmospheric  conditions.  When  this  cannot  be  done  values 
of  the  coefficient  of  refraction,  determined  for  the  region  of 
the  work,  may  be  used,  and  the  correction  d"  to  be  applied 
to  either  angle  may  be  found  by  (38)'.  Then,  either 

h  =  I  tan(a  +  d")     or     h  =  l  tan(/3  —  d") 

gives  the  difference  in  elevation  of  the  two  stations  A  and  B. 

The  best  time  for  measuring  vertical  angles  is  between 
10  A.M.  and  3  P.M.,  as  between  these  hours  the  vertical 
refraction  is  less  variable  than  either  earlier  or  later  in  the 
day.  The  less  the  distance  between  the  stations  the  less  is 
the  uncertainty  in  the  refraction,  and  the  larger  the  vertical 
angles  the  more  reliable  are  the  results.  On  account  of  the 
variability  of  refraction  and  the  inherent  inaccuracies  of  small 
angles,  elevations  found  by  vertical  angles  are  far  inferior  in 
precision  to  those  obtained  by  spirit  leveling. 

To  illustrate  the  general  method  of  procedure  let  A  and  B 
be  the  two  station  marks,  whose  horizontal  distance  apart  is 
10  352  feet.  Let  the  instru- 
ment be  set  at  A,  the  hori- 
zontal axis  of  the  telescope 
being  6. 1  feet  above  the  sta- 
tion mark,  and  pointings  be 
made  on  a  signal  b  which  is 
27.5  feet  above  the  station 
mark  B.  Let  the  mean  of  all 
the  observations  give  3°  of  15"  as  the  probable  value  of  the 
angle  of  elevation  bac.  From  (38)'  the  mean  correction  for  curv- 


106  LEVELING.  IV, 

ature  and  refraction  is  44",  so  that  the  vertical  angle  3°  of  59" 
is  the  value  to  be  used.  Then  the  difference  of  level  between 
a  and  b  is  found  by  the  use  of  the  logarithmic  tables  to  be 
566.6  feet,  and  applying  the  correction  for  height  of  instru- 

Numbers.  Logarithms. 

/     =  10352  feet  4.0150243  566.6 

bac  =  3°  07'  59"  2.7382768  21.4 

be    =  566.6  feet  2.7533031  ^  =  545.2  feet 

ment  and  signal  the  final  difference  in  elevation  between  B 
and  A  is  545.2  feet.  ,  This  is  to  be  regarded  as  liable  to  a 
probable  error  of  one  foot  or  more  on  account  of  the  uncer- 
tainties of  refraction.  To  obtain  a  better  result  the  angle  of 
depression  at  B  should  be  measured,  another  computation 
made,  and  the  mean  of  the  two  results  taken. 

It  is  customary,  when  the  distance  /  is  large,  to  reduce  the 
angle  bac  to  BAC.  Thus,  if  d  represent  the  difference 
Bb  —  Aa,  which  is  21.4  feet  in  this  case,  the  number  of 
seconds  to  be  subtracted  from  bac  is  206  265^/7,  or  426". 
Then  the  angle  BAC  is  3°  oo'  53",  and  /  tan  3°  oo'  53"  gives 

Numbers.  Logarithms. 

/          —  10352  feet  4.0150243 

BAC  =  3°  oo'  53"  £.7215257 

h         =  545-2  feet  2.7365500 

at  once  545.2  feet  as  the  difference  in  elevation  of  the  two 

station  marks. 

Prob.  39.  The  instrument  is  set  5.9  feet  above  B,  pointing  made 
on  a  signal  18.5  feet  above  A,  and  the  angle  of  depression  found  to 
be  2°  57'  30".  Compute  the  elevation  of  B  above  A. 


4O.  FUNDAMENTAL   NOTIONS.  1O7 


CHAPTER  V. 
ASTRONOMICAL    WORK. 

40.  FUNDAMENTAL  NOTIONS. 

In  a  triangulation  covering  an  area  of  some  extent  it  is 
desirable  that  the  azimuth  of  one  side  should  be  determined 
by  astronomical  work  in  order  that  the  computed  azimuths 
may  be  all  referred  to  the  true  meridian.  Rough  azimuths 
may  be  found  by  the  magnetic  needle  or  by  making  a  noon- 
mark  from  shadows  of  a  post  cast  by  the  sun.  The  method 
of  obtaining  the  meridian  with  the  solar  compass  or  transit  is 
known  to  all  surveyors,  and  it  gives  results  within  about  one 
or  two  minutes.  Azimuth  found  with  an  engineer's  transit 
from  the  sun  or  from  Polaris  furnishes  results  with  about  the 
same  precision.  For  geodetic  work,  where  an  azimuth  is 
desired  with  a  probable  error  of  only  a  few  seconds,  more 
accurate  methods  must  be  used. 

In  geodetic  triangulations  it  is  also  necessary  to  obtain  the 
astronomical  latitude  and  longitude  for  a  few  of  the  stations, 
while  those  of  the  others  are  computed  through  the  triangle 
nets.  For  the  study  of  the  figure  of  the  earth  these  astro- 
nomical observations  are  especially  important. 

A  brief  outline  of  the  field  operations  necessary  for  the 
determination  of  azimuth,  latitude,  and  longitude  will  be 
presented  in  this  Chapter.  It  is  assumed  that  the  student  is 
acquainted  with  the  fundamental  notions  regarding  the  circles 
of  the  celestial  sphere,  that  he  understands  the  method  of 


108  ASTRONOMICAL  WORK.  V. 

locating  the  position  of  a  star  by  its  right  ascension  and 
declination,  that  he  is  familiar  with  the  changes  that  occur 
throughout  the  year  in  the  declination  of  the  sun,  and  that 
he  has  a  knowledge  of  the  different  ways  of  measuring  time. 
In  short,  he  should  have  had  a  good  course  in  descriptive 
astronomy. 

In  geography  the  latitude  of  a  place  is  its  angular  distance 
from  the  terrestrial  equator,  and  in  astronomy  it  is  the 
angular  distance  of  the  zenith  of  the  place  from  the  celestial 
equator.  Thus  astronomical  latitude  is  determined  with 
reference  to  a  vertical  line  at  the  point  of  observation.  Since 
the  horizon  plane  is  perpendicular  to  this  line  it  follows  that 
latitude  is  the  angular  distance  of  the  pole  above  the  horizon. 

Thus,  if  A  be  any  place,  repre- 
sented in  the  figure  by  a  point  at 
the  center  of  the  celestial  sphere, 
the  vertical  line  AZ  determines  the 
zenith  Z.  Let  P  be  the  celestial 
pole  and  N  the  north  point  of  the 
horizon,  then  the  angle  PAN  or  the 
arc  PN  is  the  latitude  of  the  place. 
In  all  questions  relating  to  lati- 
tudes it  is  well  for  the  student  to  remember  that  one  minute 
corresponds  approximately  to  one  nautical  mile  on  the  earth's 
surface,  and  one  second  to  about  101  feet  (Art.  53). 

The  above  figure  sets  forth  several  of  the  fundamental 
notions  of  astronomy.  The  horizon  of  the  place  A  is  shown 
by  a  circle  joining  TV  and  C,  the  meridian  of  the  place  by  the 
circle  NPZ,  and  its  co-latitude  by  the  arc  PZ.  If  5  be  the 
sun  or  a  star  QS  is  its  declination  and  PS  its  co-declination, 
CS  is  its  altitude  and  ZS  its  co-altitude  or  zenith  distance. 
In  the  spherical  triangle  SZP  the  angle  at  P  is  the  hour-angle 
of  the  sun  or  star,  while  the  angle  at  Z  gives  its  angular  dis- 
tance from  the  meridian,  that  is,  its  azimuth.  In  the  summer 
the  sun's  apparent  daily  path  lies  north  of  the  equator,  at  the 


40.  FUNDAMENTAL   NOTIONS.  IOO, 

equinoxes  it  lies  on  the  equator,  and  in  the  winter  it  lies 
south  of  the  equator. 

Azimuths  and  hour-angles  in  astronomy  are  generally 
measured  from  the  south  around  through  the  west  from  o°  to 
360°,  like  azimuth  in  geodesy.  It  is,  however,  sometimes 
convenient  to  give  them  negative  values;  thus  in  the  figure, 
if  5  represents  the  sun  about  10  o'clock  in  the  morning,  its 
hour-angle  ZPS  is  330°  or  —  30°,  and  its  azimuth  MZS  is 
about  310°  or  —  50°.  Azimuths  of  circumpolar  stars  are 
sometimes  estimated  from  the  north  toward  the  east  and  180° 
is  then  to  be  added  to  give  geodetic  azimuths. 

The  methods  to  be  here  presented  are  those  that  can  be 
carried  out  in  the  field  with  an  engineer's  transit  or  with  a 
sextant.  The  results  are  not  as  precise  as  those  derived  with 
the  instruments  of  an  observatory  or  with  portable  astro- 
nomical instruments,  but  the  fundamental  principles  and 
methods  are  the  same  and  hence  this  Chapter  may  serve  as 
an  introduction  to  the  practical  field  operations  of  geodetic 
astronomy.  Azimuth  is  the  most  important  problem  for  the 
civil  engineer  and  it  will  be  presented  first,  assuming  that  the 
latitude  and  longitude  of  the  place  have  been  found  from  a 
map  and  that  standard  time  is  given  by  a  watch.  Afterwards 
it  will  be  shown  how  latitude,  time,  and  longitude  can  be 
determined. 

In  most  of  the  work  of  practical  astronomy  an  almanac 
must  be  at  hand  to  furnish  the  declinations,  right  ascensions, 
and  other  data  that  are  needed  in  the  computations.  The 
American  Nautical  Almanac,  which  is  published  by  the 
Bureau  of  Equipment  of  the  U.  S.  Navy  and  can  be  had 
through  any  bookseller  for  fifty  cents,  will  give  all  the  data 
required  for  the  work  of  this  chapter. 

Prob.  40.  The  declinations  of  the  sun  at  Greenwich  mean  noon 
on  March  20  and  21,  1900,  were  — o°  13'  28". 4  and  +  o°  10' 
1 3".  4.  When,  in  Eastern  standard  time,  did  the  vernal  equinox 
occur  ? 


IIO  ASTRONOMICAL   WORK.  V, 


41.  AZIMUTH  BY  THE  SOLAR  TRANSIT. 

With  a  transit  having  a  solar  attachment  the  azimuth  of  a 
line  can  be  found  by  observing  the  sun  at  any  time  except 
between  II  A.M.  and  I  P.M.,  the  most  favorable  hours  being 
generally  from  9  to  10  A.M.  and  from  2  to  3  P.M.  Such  an 
attachment  can  be  placed  upon  any  transit  at  a  cost  of  about 
fifty  dollars.  Accompanying  it  is  a  pamphlet  giving  full 
directions  for  use  and  adjustment,  together  with  tables  of  the 
declination  of  the  sun  for  Greenwich  noon  for  each  day  of  the 
year.  Both  the  transit  and  the  solar  attachment  should  be 
in  correct  adjustment  in  order  to  do  good  work. 

Let  the  upper  part  of  the  figure  represent  a  section  of  the 
celestial  sphere  in  the  plane  of  the  meridian,  A7"  and  M  being 
the  north  and  south  points  of  the  horizon,  P  the  pole,  Z  the 

zenith,  Q  the  celestial  equator,  and 
5  the  place  of  the  sun  at  noon.  Let 
A  be  the  point  where  the  instrument 
is  set,  which  may  be  regarded  as  the 
center  of  the  celestial  sphere.  Then 
the  angle  QAZ,  or  its  equal  PAN,  is 
the  latitude  of  the  place  of  observa- 
tion. The  angle  QAS  is  the  declina- 
tion of  the  sun,  which  is  positive  when 
the  sun  is  north  of  the  equator  and  negative  when  it  is  south 
of  the  equator.  The  lower  part  of  the  figure  is  a  plan,  A 
being  the  place  of  the  instrument,  NM  the  true  meridian,  W 
and  E  the  west  and  east  directions,  AS  the  direction  of  the 
sun  about  10  o'clock  in  the  morning,  and  AB  a  line  whose 
azimuth  is  required  to  be  found. 

Let  ab  represent  the  telescope  of  the  transit,  it  being 
represented  as  in  the  meridian  and  elevated  so  as  to  point  to 
the  celestial  equator;  this  will  be  the  case  when  the  angle 
of  elevation  MAQ  is  equal  to  the  co-latitude  or  when 
MAQ  =  90°  —  QAZ.  Let  cd  represent  the  telescope  of  the 


AZIMUTH    BY   THE   SOLAR   TRANSIT. 


Ill 


solar  attachment  pointing  toward  the  sun;  then  the  vertical 
angle  between  ab  and  cd  is  -equal  to  the  declination  of  the 
sun  QAS.  In  this  position^  the  solar  attachment  is  like  an 
equatorial  telescope'  its  axis  pointing  to  the  pole  P,  and  as 
the  sun  moves  the  telescope  cd  can  be  made  to  follow  it  by 
simply  turning  it  on  its  axis. 

Before  beginning  the  work  a  list  of  hourly  declination 
settings  is  to  be  prepared  by  help  of  the  table  of  declinations 
which  is  annually  furnished  by  the  maker  of  the  instrument. 
This  table  also  gives  the  corrections  to  be  applied  for  refrac- 
tion, these  always  being  added  to  the  true  declinations, 
because  refraction  increases  the  true  altitude  of  the  sun. 
For  example,  let  it  be  required  to  prepare  the  declination 
settings  for  the  afternoon  of  September  16,  1899,  for  any 
place  where  Eastern  standard  time  is  used.  The  table  gives 
-[-  2°  37'  44". 4  as  the  declination  of  the  sun  at  Greenwich 
mean  noon  of  that  day  and  57". 91  as  the  hourly  decrease  in 
declination.  At  7  A.M.  of  Eastern  standard  time  the  declina- 
tion is  hence  +  2°  37'  44". 4,  at  5  P.M.  it  is  +  2°  37'  44". 4 
—  10  X  57". 91  —  +  2°  28'  05". 3,  and  at  4  P.M.  it  is  +  2° 
28'  os".3  +  57".9  =  +  2°  29'  03//.2.  Thus  the  declination 
for  each  hour  is  found  and  placed  in  the  second  column.  In 

DECLINATIONS  FOR  SEPTEMBER  16,  1899. 


Hour. 

Declination. 

Refraction 
Correction. 

Declination 
Setting. 

Remarks. 

P.M. 

I 
2 

+  2°  31'  57" 
+  2     30    59 

+  o'    48" 

-f  o     54 

+  2°    32'  45" 
-f  2     31    53 

For  Eastern 
Standard  time. 

3 

+  2       30     01 

+  i     05 

+  2      31      06 

4 

+  2       29     03 

+  i     32 

+  2     30    35 

Lat.  40°  36'. 

5 

4-2       28     05 

4-2     51 

4-2      30     56 

the  third  column  are  placed  the  refraction  corrections  as  given 
in  the  table,  and  the  fourth  column  contains  the  final  declina- 


112 


ASTRONOMICAL  WORK. 


V. 


tions  to  be  set  off  on  the  vertical  arc  as  closely  as  its  gradua- 
tion will  allow.  The  refraction  correction  is  always  additive, 
and  hence  if  the  declination  is  south  or  negative  its  numerical 
value  is  to  be  decreased,  as  the  example  for  December  2, 
1899,  shows;  for  that  day  the  table  gives  the  declination  at 
Greenwich  mean  noon  as  —21°  58'  48". 3  and  the  hourly 
change  as  22".2O. 

DECLINATIONS  FOR  DECEMBER  2,  1899. 


Hour. 
A.M. 

Declination. 

Refraction 
Correction. 

Declination 
Setting. 

Remarks. 

8 
9 

—  21°    59'    10" 
-  21     59    33 

+  6'    01" 
+  2     59 

-  21°   S3'  Q9" 
-  21     56    34 

For  Eastern 
Standard  time. 

IO 

-  21     59    55 

+  2      II 

-  21     57    44 

Lat.  40°  36'. 

II 

—   22      00      17 

+  i     54 

—  21     58     23 

After  this  list  is  made  out  the  observer  sets  up  the  transit 
over  the  point  A  in  order  to  find  the  azimuth  of  a  line  AB. 
The  telescope  is  leveled  by  the  attached  bubble  and  pointed 
in  a  southerly  direction.  The  declination  setting  for  the 
hour  is  next  laid  off  on  the  vertical  arc,  depressing  the  object 
glass  if  the  declination  is  positive  and  elevating  it  if  the 
declination  is  negative.  The  telescope  of  the  solar  attach- 
ment is  then  leveled  by  means  of  its  own  bubble,  and  thus 
the  angle  between  the  two  telescopes  is  the  same  as  the 
apparent  declination,  or  the  angle  QAS  in  the  above  figure. 
Both  telescopes  are  then  elevated  until  the  vertical  arc  reads 
an  angle  equal  to  the  co-latitude  of  the  place,  or  the  angle 
MAQ.  The  solar  attachment  is  next  turned  on  its  axis  and 
the  limb  of  the  transit  upon  its  axis  until  the  sun  is  seen 
inscribed  in  the  square  formed  by  the  four  extreme  cross-hairs 
in  the  focus  of  the  solar  telescope.  When  this  is  the  case 
the  transit  telescope  is  in  the  plane  of  the  meridian,  and  if 
desired  a  point  may  be  set  out  in  the  line  AS  to  mark  that 
meridian. 


41- 


AZIMUTH    BY   THE   SOLAR   TRANSIT. 


It  will  be  better,  however,  to  read  both  verniers  on  the 
horizontal  circle,  then  turn  the  alidade  and  sight  on  B,  and 
read  both  verniers  again.  The  angle  MAB  has  thus  been 
measured  and,  for  the  position  in  the  figure,  this  is  to  be 
subtracted  from  360°  to  give  the  geodetic  azimuth  of  AB. 

FIELD  NOTES  FOR  AZIMUTH  OF  AB. 


,Time. 
October  28, 
1895. 

Reading  on  Meridian. 

Reading  on  Line  AB. 

Angle  MAB. 

Remarks, 

9:15  A.M. 

2O°    19' 

oo" 

30" 

l82°    27" 

30" 

30" 

l62°    08'     15" 

R.  Doe, 

9:30 

80    oo 

15 

15 

242      08 

30 

30 

162     09     oo 

Observer. 

9:45 

140     59 

30 

15 

303      08 

45 

15 

l62      09      08 

3:15   P.M. 
3--30 

200      01 
260       12 

60 

45 

45 
30 

2      09 
62      22 

45 
15 

30 
30 

162     07     45 
162     09     45 

n  =  32" 

3:45 

32O      O6 

oo 

oo 

122       13 

45 

60 

162     07     53 

r    -  13" 

Mean  =  162°  08'  38" 

Azimuth  of  AB  =  197°  51'  22" 

The  above  form  of  field  notes  shows  six  observations  made  in 
this  manner,  and  from  their  mean  is  found  197°  51'  22"  for 
the  azimuth  of  AB.  The  probable  error  of  this  mean  is 
determined  by  Art.  9  to  be  about  13",  that  of  a  single  obser- 
vation being  32".  This  degree  of  precision  is  greater  than 
can  be  generally  attained  by  azimuth  observations  with  the 
solar  attachment,  unless  the  observer  has  had  considerable 
experience;  nevertheless  by  a  moderate  amount  of  practice  it 
is  easy  to  determine  an  azimuth  with  a  probable  error  of  less 
than  one-half  a  minute,  both  morning  and  afternoon  observa- 
tions being  taken. 

Prob.  41.  Take  several  observations  of  the  azimuth  of  a  line  by 
the  solar  transit,  and  find  the  probable  error  of  their  mean.  Explain 
how  the  solar  transit  differs  from  the  solar  compass  and  state  the 
advantages  of  the  former  over  the  latter. 


114  ASTRONOMICAL   WORK.  V. 

'        %       v. 

42.  AZIMUTH  BY  AN  ALTITUDE  OF  THE  SUN. 

The  azimuth  of  a  given  line  may  be  determined  by  taking 
the  altitude  of  the  sun  with  an  engineer's  transit  having  a 
good  vertical  circle  and  reading  the  horizontal  angle  between 
the  sun  and  the  line.  The  latitude  of  the  place  must  be 
known,  and  a  nautical  almanac  must  be  at  hand  for  finding 
the  declination  of  the  sun  at  the  moment  of  observation. 

In  the  figure  let  A  represent  the  center  of  the  celestial 
sphere,  P  the  pole,  Z  the  zenith,  N  the  north  point  of  the 
horizon,  and  5  the  position  of  the  sun  at  the  moment  of 
observation.  Then,  in  the  spherical  triangle  PZS  the  angle 
Z  is  the  azimuth  of  the  sun  measured  from  the  north  around 
through  the  east,  and  this  is  the  same  as  the  horizontal  angle 
NAC.  Let  AB  be  the  line  whose  azimuth  is  to  be  found; 
then  if  the  horizontal  angle  CAB  be  measured  its  azimuth  is 
known  as  soon  as  Z  has  been  found. 

In  the  figure  CS  is  the  altitude  of  the  sun,  and  SZ  is  the 

complement  of  that  altitude  or  the 
zenith  distance  of  the  sun;  let  the 
latter  be  represented  by  2.  Let  0 
be  the  latitude  of  the  place,  or  the 
arc  NP.  Let  8  be  the  declination 
of  the  sun  or  the  arc  QS.  Then 
in  the  spherical  triangle  PZS,  the 
three  sides  are  known,  and  hence 
the  angle  Z  can  be  found  from  the 
equation  of  spherical  trigonometry 

sintf  =  cos-s-  sin0  -f-  sin-8"  cos0  cosZ. 

For  accuracy  of  computation,  it  is  best  to  put  this  into 
another  form;  thus  by  writing  cosZ  =  I  —  2  sin^Z  a  value 
is  found  for  sin-J2T,  and  by  writing  cosZ  =  —  i  -f-  2  cos'JZ  a 
value  is  found  for  cos^Z;  then  dividing  the  first  value  by  the 
second  there  results 


42.  AZIMUTH    BY  AN   ALTITUDE   OF  THE   SUN.  11$ 


_        /cosK*  +  0  +  <?)    sinjQ  +  0  -  tf)      . 
:  V  costf*  -,  0  -  6)    sini(*  -  0  +  6)' 

from  which  Z  is  t<*  be  computed.  In  the  figure  5  denotes 
the  place  of  the  sun  in  the  summer  half-year  when  6  is  posi- 
tive, and'S'  its  place  in  the  winter  half-year  when  d  is  nega- 
tive. If  the  observation  is  taken  in  the  forenoon  the  geodetic 
azimuth  (Art.  13)  of  the  sun  is  180°  -j-  Z,  if  in  the  afternoon 
it  is  1 80°  —  Z. 

The  transit  having  been  put  into  thorough  adjustment  it 
is  set  up  at  A,  the  end  of  the  line  AB  whose  azimuth  is  to 
be  determined.  The  horizontal  limb  is  clamped,  a  reading  of 
the  horizontal  circle  taken,  and  then  the  telescope  pointed 
at  B.  The  alidade  is  then  undamped  and  the  telescope 
pointed  at  the  sun,  the  objective  and  eye-piece  being  so 
focused  that  the  shadow  of  the  cross-wires  and  the  image  of 
the  sun  may  be  plainly  seen  upon  a  piece  of  white  paper  held 
behind  the  eyepiece.  The  cross-wires  should  be  made 
tangent  to  the  image  on  its  lower  and  right-hand  sides  and 
the  horizontal  and  vertical  circles  be  read;  next  they  should 
be  made  tangent  to  the  image  in  its  upper  and  left-hand  sides 
and  the  two  circles  be  read  again.  If  the  transit  has  a  full 
vertical  circle,  which  is  necessary  for  the  best  work,  observa- 
tions should  be  taken  both  in  the  direct  and  reverse  position 
of  the  telescope. 

The  following  record  will  illustrate  the  method  of  making 
the  measurements  and  obtaining  the  data  for  computation. 
The  declination  8  for  8:43  A.M.,  Eastern  standard  time,  of 
the  day  of  observation  is  taken  from  a  nautical  almanac. 
The  mean  apparent  altitude  is  43°  58'  22" ,  and  this  being 
corrected  for  parallax  and  refraction  the  zenith  distance  z  is 
found  to  be  46°  02'  52" ' .  By  computation  from  the  formula 
the  azimuth  Z,  or  the  angle  NACy  is  found  to  be  101°  45' 
36",  whence  the  geodetic  azimuth  of  the  sun  at  the  middle 
of  the  observation  is  281°  45'  $6".  Subtracting  from  this 


n6 


ASTRONOMICAL  WORK. 


V. 


the  mean  horizontal  angle  BA  C  the  geodetic  azimuth  of  the 
line  AB  is  found  to  be  216°  44'  06". 


AZIMUTH  OF  AB  BY  THE  SUN. 


Time. 
May  19, 
1897. 

Tel. 

Vertical  Angle. 
CAS. 

Horizontal 
Angle. 
BAC. 

Data  and  Results. 

A.M. 

Wires  tan 

gent  to  lower 

0  =  40°  36'  27" 

and  right 

sides. 

8  at  7A.M.  =  19°  53'  10" 

8"  40» 

D 

43°  09'   oo" 

64°  48'   oo" 

55 

42 

R 

43     35     30 

65     10     30 

8  =  19°  54'  05" 

Wires  tan 

gent  to  upper 

App.  altitude  =  43°  58'  22" 

and  left 

sides. 

Par.(+o6"),Ref.(-6o")  -54 

8     44 

R 

44°  21'    oo" 

64°  52'    30" 

True  altitude  =  43   57    28 

46 

D 

44     48     oo 

65     15     oo 

90 

/       // 

z  —     40    02    32 

Means  = 

43°  58'    22" 

65°  01     30" 

Z  =  101°  45'  36" 
180 

65    01    30 

Azimuth  AB  =  216°  44'  06" 

The  correction  for  parallax  of  the  sun  is  less  than  9"  and 
it  is  always  added  to  the  apparent  altitude.  For  an  altitude 
of  20°  the  parallax  correction  is  8",  for  40°  it  is  7",  and  fcr 
60°  it  is  6" ' .  In  precise  work  the  value  of  this  correction 
may  be  found  by  multiplying  8". 9  by  the  cosine  of  the 
apparent  altitude  of  the  sun. 

The  refraction  correction  is  taken  from  Table  I  at  the  end 
of  this  volume;  it  is  always  subtracted  from  the  apparent 
altitude,  since  the  effect  of  refraction  is  to  render  the  apparent 
altitude  greater  than  the  true  altitude. 

The  probable  error  of  a  single  azimuth  observation  made 
by  this  method  is  usually  one  or  two  minutes;  to  secure  a 
precise  result  several  observations  should  be  made  both  in 
the  forenoon  and  afternoon  and  the  mean  of  the  computed 
values  be  taken.  The  best  time  for  the  work  is  when  the  sun 


43-  AZIMUTH   BY  POLARIS  AT   ELONGATION.  1 1/ 

is  near  the  prime  vertical,  that  is,  nearly  east  or  west.  Near 
the  noon  hour  the  methotl  is  .of  no  value,  since  then  a  small 
error  in  z  causes  ajarge  erfbr  in  Z;  moreover  when  the  sun 
i?  on  the  meridian  z  =  0  —  d  identically. 

More  precise  results  can  be  obtained  by  using  a  star  instead 
)f  the  sun.  In  this^  case  the  observer  looks  at  the  image  of 
the  star  in  the  telescope  and  brings  it  to  coincide  with  the 
intersection  of  the  cross-wires.  The  star  is  usually  not  bright 
enough  to  illuminate  the  cross-wires  and  hence  it  is  necessary 
to  throw  light  into  the  objective  end  of  the  telescope  by 
means  of  a  lamp  held  about  a  foot  or  two  on  one  side  of  it. 
The  signal  at  the  end  of  the  line  AB  must  also  be  illuminated. 
The  declination  of  the  star  is  taken  from  the  nautical  almanac 
with  less  trouble  than  that  of  the  sun  as  its  daily  change  is 
inappreciable.  The  apparent  altitude  needs  no  correction  for 
parallax,  but  the  refraction  correction  is  to  be  applied.  With 
these  exceptions  the  method  of  observation  and  computation 
is  identical  with  that  above  explained.  In  the  winter  season, 
when  the  sun  cannot  be  observed  near  the  prime  vertical,  a 
star  favorable  for  observation  can  always  be  found. 

Prob.  42.  In  latitude  38°  53'  18",  when  the  declination  of  a  star 
was  -+-  13°  55'  33",  the  apparent  observed  altitude  was  28°  42' 58". 
Find  the  corrected  zenith  distance  and  compute  the  azimuth  of  the 
star. 

43.  AZIMUTH  BY  POLARIS  AT  ELONGATION. 

When  Polaris  is  approaching  its  eastern  or  western  elonga- 
tion it  may  be  easily  followed  by  the  vertical  wire  in  the 
telescope  of  an  engineer's  transit,  and  when  its  motion  in 
azimuth  ceases  a  horizontal  angle  may  be  read  between  its 
direction  and  that  of  a  given  line.  The  azimuth  of  Polaris 
at  elongation  being  known  that  of  the  line  is  immediately 
found, 

In  the  figure  let  Z  be  the  zenith,  P  the  pole,  N  the  north 
point  of  the  horizon,  HH  the  horizon  itseh,  and  E  and 


ASTRONOMICAL   WORK, 


V. 


positions  of  Polaris  at  the  eastern  and  western  elongations. 
PN  is  the  latitude  of  the  place  of  observation  and  hence  PZ 
is  the  co-latitude  90°  —  0;  PE  or  PW  is  the  co-declination 
of  Polaris  at  elongation  or  90°  —  #;  the  angle  PZE  or  PZW 
is  the  azimuth  of  Polaris  at  elongation  measured  eastward  or 
westward  from  north.  Let  this  azimuth  be  called  Z;  then, 
as  the  spherical  triangles  are  right-angled  at  E  and  W, 


sinZ  =  cos#/cos0, 


(43) 


from  which  Z  can  be  found  for  any  given  latitude  when  d  has 
been  taken  from  the  nautical  almanac.  The  declination  of 
Polaris  is  slowly  increasing  at  the  rate  of  about  19"  per  year, 
its  value  being  88°  46'  26" .6  for  Jan.  I,  1900,  and  88°  46' 
45". 4  for  Jan.  i,  1901. 

The  approximate,  times  of  the  elongations  of  Polaris  for 
each  month  in  the  year  are  given  in  surveyor's  handbooks 
and  need  not  be  repeated  here.  Half 
an  hour  before  the  time  the  observer 
sets  the  transit  at  A  and  places  a 
signal,  illuminated  if  necessary,  at  B. 
The  horizontal  circle  is  read,  the  lower 
limb  being  clamped,  and  the  telescope 
is  pointed  at  B\  the  alidade  is  then 
undamped,  the  telescope  pointed  at 
Polaris,  which  is  followed  until  it  reaches 
its  elongation,  and  then  the  horizontal 
circle  is  read  again.  Thus  on  August 
15,  1899,  at  about  9:50  P.M.  local  time, 
an  eastern  elongation  occurred  and  an 
observer  in  latitude  40°  $6'  took  the 
reading  87°  09'  30"  when  the  pointing 
was  made  on  B  and  74°  04'  oo"  on 

Polaris;  the  horizontal  angle  HAB  is  hence  13°  05'  30". 
From  the  nautical  almanac  the  value  of  d  is  88°  46'  19"  and 
then  by  the  formula  the  value  of  Z  is  i°  37'  04".  Thus  for 


43-  AZIMUTH    BY    POLARIS   AT    ELONGATION.  1 19 

this  case,  as  shown  in  the  figure,  the  direction  of  AB  is 
14°  42'  34"  to  the  eastward  of  the  meridian  and  accordingly 
its  geodetic  azimuth  is  193°  42'  34". 

By  the  above  method  only  one  reading  of  the  horizontal 
circle  can  be  taken  on  Polaris,  and  hence  there  is  no  oppor- 
tunity to  eliminate'  the  various  sources  of  error  of  the  transit. 
It  is,  however,  possible  to  measure  a  number  of  angles  before 
and  after  elongation  and  apply  to  each  a  correction  to  reduce 
it  to  elongation.  For  this  purpose  the  time  of  elongation 
should  be  known  and  this  can  be  found  in  local  time  within 
less  than  half  a  minute  by  the  tables  in  the  Handbook  for 
Surveyors.  Five  pointings  on  Polaris  may  then  be  made 
during  the  quarter-hour  preceding  elongation  and  five  during 
the  quarter-hour  following  elongation.  A  good  plan  is  to 
take  these  exactly  at  the  beginning  of  three-minute  intervals, 
then  to  read  the  verniers,  turn  to  the  signal  or  mark  at  B  and 
read  again.  Half  the  angles  are  read  with  the  telescope  in 
the  direct  position  and  half  with  it  in  the  reverse  position. 
The  readings  are  distributed  over  the  circle  by  making  each 
one  about  20°  greater  than  the  preceding.  The  following 
form  of  field  notes  will  render  clear  the  method  of  conducting 
the  work.  The  eastern  elongation  was  to  occur  at  9:49  P.M. 
in  the  time  indicated  by  the  watch  of  the  observer,  and  it 
was  arranged  to  take  the  five  pointings  before  elongation  14, 
n,  8,  5,  and  2  minutes  earlier,  while  those  following  were 
taken  I,  4,  7,  10,  and  13  minutes  later,  as  shown  in  the 
second  column.  Thus  ten  horizontal  angles  were  measured 
between  the  star  and  the  illuminated  mark  at  B.  To  reduce 
these  to  elongation  a  correction  c  is  to  be  subtracted  from 
each,  this  being  computed  from  the  formula  c  =  o.o/w2, 
where  n  is  the  number  of  minutes  of  time  preceding  or  fol- 
lowing elongation.  These  values  of  c  are  seen  in  the  last 
column,  and  the  mean  corrected  angle  HAB  is  found  to  be 
13°  05'  33",  from  which  the  final  geodetic  azimuth  of  AB  is 
194°  42'  37". 


120 


ASTRONOMICAL  WORK. 
AZIMUTH  BY  POLARIS.     AUGUST  16,  1899. 


V. 


Time. 

n 

Tel. 

Reading  on  Polaris. 

Reading  on  Mark  B. 

Horizontal 

c 

P.M. 

Angle. 

9=35 

+   "4 

D 

o     17 

00 

30 

13      22 

40 

60 

13    05    40 

~  13 

38 

+  H 

R 

20      42 

30 

50 

33     48 

45 

50 

13   05    47 

-    8 

41 

+    8 

R 

40      03 

IO 

20 

53     08 

40 

50 

13    05    30 

—    4 

44 

+    5 

D 

60      40 

40 

50 

73     46 

00 

IO 

13    05     20 

—      2 

47 

+     2 

D 

80    oo 

IO 

IO 

93     05 

50 

40 

13   05    35 

0 

50 

—     I 

R 

100    31 

20 

05 

113     36 

40 

40 

13    05     27 

o 

53 

-    4 

R 

1  20     04 

15 

IO 

133     09 

60 

40 

13   05    37 

—     I 

56 

-     7 

D 

140    17 

50 

50 

153     23 

30 

30 

13   05   40 

-    3 

59 

—    10 

r 

160     14 

50 

50 

173     JO 

50 

60 

13    06   05 

-    7 

10:02 

-  13 

R 

180     25 

10 

00 

193     30 

50 

40 

13    05    40 

—    12 

Time  of  elongation  9:49  P.M.         Mean  corrected  HAB  —     13°  05'  33" 

d  =  88°  46'  19"                                                            180  +  Z  =  181    37  04" 

0~4O    36  oo                                                Azimuth  of  AB  =  194°  42'  37" 

Z  =     i    37  04                                        J.  Doe,  observer  and  computer. 

For  work  with  an  engineer's  transit  the  corrections  can  be 
found  close  enough  south  of  latitude  50°  by  the  approximate 
rule  c  =  O.O7/Z2,  but  for  observations  with  a  theodolite,  where 
tenths  of  seconds  are  to  be  used,  the  more  accurate  formula 
given  in  treatises  on  practical  astronomy  should  be  employed. 

The  precision  of  this  method  depends  almost  wholly  upon 
that  of  the  pointings  and  readings.  If  the  declination  of 
Polaris  be  taken  from  the  nautical  almanac  by  interpolating 
for  the  day  of  observation  no  error  can  arise  from  this  source. 
The  error  in  the  computed  Z  due  to  an  error  of  one  minute 
in  the  latitude  will  range  from  i"  at  latitude  30°  to  about  2" 
at  latitude  48°.  The  probable  error  of  a  single  angle  meas- 
urement may  range  from  5"  to  40",  depending  upon  the  skill 
of  the  observer  and  the  kind  of  transit  or  theodolite  used,  and 
accordingly  the  probable  error  of  an  azimuth  found  from  a 


44.  AZIMUTH   BY   POLARIS   AT  ANY   HOUR-ANGLE.  121 

series  of  ten  angles  may  range  from  2"  to  15".  The  precision 
of  the  series  above  given  is  considerably  greater  than  can  be 
generally  secured  by  an  engineer's  transit  reading  to  half- 
minutes. 

This  method  is  advantageous  from  its  simplicity,  but  dis- 
advantageous because  at  the  utmost  only  two  observations 
can  be  taken  in  twenty-four  hours.  For  a  single  reading 
taken  exactly  at  elongation  the  time  need  not  be  known 
further  than  to  be  sure  of  being  ready  a  few  minutes  before 
it  occurs.  For  several  readings  it  should  be  known  within 
half  a  minute,  so  that  the  times  of  the  pointings  may  be 
arranged  in  advance. 

Prob.  43.  Show  that  the  error  in  a  computed  azimuth  due  to  an 
error  in  latitude  increases  with  the  tangent  of  the  latitude  ;  or  if  d(f> 
is  the  error  in  latitude  show  that  dZ  =  —  tanz 


44.  AZIMUTH  BY  POLARIS  AT  ANY  HOUR-ANGLE. 

Polaris  or  any  other  circumpolar  star  may  be  used  at  any 
position  for  the  determination  of  azimuth,  if  the  observer's 
watch  indicates  correct  time,  either  local  or  standard,  and  if 
the  latitude  and  longitude  of  the  place  are  known.  From  the 
time  of  observation  and  the  data  given  in  the  nautical  almanac 
the  hour-angle  of  the  star  is  to  be  found  and  the  solution  of 
a  spherical  triangle  then  gives  the  azimuth  of  the  star. 

Let  Z  be  the  zenith,  P  the  pole,  5  the  place  of  the  star, 
and  N  the  north  point  of  the  horizon.  In  the  spherical 
triangle  PZS  the  angle  Z  is  the  azimuth  of  the  star  east  of 
the  meridian,  and  the  angle  at  P  is  its  hour-angle  minus  180°; 
the  side  ZP  is  the  co-latitude  90°  —  0,  the  side  PS  is  the  co- 
declination  90°  —  6.  Let  /  denote  the  hour-angle  of  the  star, 
that  is  the  obtuse  spherical  angle  NPS\  then  the  solution  for 
the  angle  Z  gives 

sin/ 
cos/  -  cos0  taneT  (44) 


122 


ASTRONOMICAL  WORK. 


V. 


from  which  Z  is  to  be  computed  after  the  hour-angle  t  has 
been  determined. 

The  field  operations  may  be  conducted  exactly  like  those 
explained  in  the  last  Article.  Another  method  of  observation 
preferred  by  many  observers  is  illustrated  in  the  notes  be- 
low. Pointing  is  first  made  on  the  mark 
at  B  and  the  horizontal  circle  is  read; 
then  four  pointings  and  readings  are 
made  on  the  star,  two  with  the  telescope 
in  the  direct  position  and  two  with  it 
in  the  reverse  position ;  finally  a  point- 
ing and  reading  on  the  mark  is  taken 
again;  each  reading  is  of  course  the 
mean  of  the  two  verniers.  The  time  as 
indicated  by  the  watch  must  be  noted 
for  each  pointing  on  the  star,  and  the 
mean  of  these  times  is  that  to  be  used 
to  find  the  hour-angle  /.  The  process 
of  finding  t  by  the  help  of  the  nautical 
almanac  is  shown  in  the  lower  part  of 
the  table.  Then  from  the  formula  tanZ  is  found  to  be 
negative  and  hence  the  star  was  west  of  the  meridian; 
accordingly  Z  is  —  o°  59'  06". 3,  and  finally  the  geodetic 
azimuth  of  the  line  AB  is  175°  57'  47". 6,  the  probable  error 
of  which  may  be  estimated  at  10"  or  15".  Making  a  number 
of  observations  on  different  parts  of  the  circle  and  taking 
their  mean,  a  fair  determination  of  azimuth  may  be  obtained 
by  one  night's  work'. 

To  secure  the  elimination  of  the  instrumental  errors  of  the 
transit  more  completely,  one  half  of  the  pointings  on  the  star 
may  be  made  by  looking  at  its  reflection  in  a  dish  of  mercury 
placed  near  the  objective  end  of  the  telescope.  When  a 
geodetic  theodolite  is  used  corrections  for  the  error  of  level 
in  the  telescope  standards  are  to  be  applied,  unless  this  be 
eliminated  by  taking  half  the  observations  in  a  mercury  hori- 


44- 


AZIMUTH    BY   POLARIS   AT  ANY    HOUR-ANGLE. 


123 


zon.  If  a  sidereal  chronometer  is  at  hand  the  time  should 
be  noted  by  it,  as  thus  the  reduction  of  local  mean  solar  time 
to  sidereal  time  is  avoided.^ 

AZIMUTH  BY  POLARIS,  APRIL  29,  1897. 


Watch  Time 

P.M. 

Tel. 

Readings  on  Mark. 

Readings  on  Star. 

Data. 

D 

25°    II'    20" 

-0  =  40°   36'   24" 

R 

25      II      20 

A.  =  75°    22'   50" 

8h  i6m  30s 

R 

30°    14'    30" 

S  =-.  88°  45'  35"-6 

8    18     30 

D 

30      13      05 

a  =     ih  20m  19".  2 

8    20     30 

D 

30      II      50 

8    22     30 

R 
R 

25     ii     oo 

30    10    40 

Watch  93s.  3  faster 
than  Eastern 
standard  time. 

D 

25     ii     10 

J.  Doe,  observer. 

8h  I9m  30" 

25°  ii'    I2/X5. 

30°  12'    3i".2 

_  oi     33.3  =  Watch  error. 

30°  12'  31".  2 

8h      I7m  s68.7  =  Eastern  standard  time. 
—  oi      3^  .3  =  Longitude  correction. 

25    ii    12  .5 

BAH1  =      5    oi   18  .7 
Z=  —o   59  06  .3 

8       16      25  .4 

+  01      21    .7 

=  Loc 

al  mean  solar  time. 

luction  to  sidereal 
nterval. 

=  Rec 
i 

BAN—      4   02  12  .4 

2         30      48    .5 

+  oo    49  .4 

=  Sidereal  time  Greenwich 
mean  noon. 

=  Longitude  correction. 
=  Sidereal  time. 

1  80 

Azimuth  AB  =  175°  57'  47".  6 
R.  Roe,  computer. 

ioh     49m  25'  .0 

I         20      19    .2 

=  Right  Ascension  of  Polaris. 
=  Hour-angle  in  time. 

9U      29m  05".  8 

142°      16'    27".  3  =/ 

Any  circumpolar  star  may  be  used  by  this  method,  but 
preference  is  generally  given  to  Polaris  as  it  is  of  second 
magnitude  and  easily  identified.  Other  stars  sometimes  used 
are  d  Ursae  Minoris  and  51  Cephei,  which  are  of  fifth  magni- 
tude and  hence  not  so  easily  located  as  Polaris. 


124 


ASTRONOMICAL   WORK. 


V. 


Although  theoretically  the  observation  may  be  taken  at 
any  time,  yet  a  discussion  of  equation  (44)  will  show  that  the 
conditions  most  favorable  to  precision  occur  when  t  is  either 
about  90°  or  270°,  that  is  when  the  star  is  near  elongation. 
When  the  star  is  at  either  the  upper  or  lower  culmination 
small  errors  in  0  and  /  may  produce  large  errors  in  Z,  and 
hence  a  star  should  not  be  observed  when  near  its  meridian 
passage.  Errors  due  to  either  0  or  6  may  be  eliminated  by 
observing  the  star  at  symmetrical  positions  east  and  west  of 
the  meridian,  and  taking  the  mean  of  the  two  computed 
results. 

Prob.  44.  If  an  error  of  15  seconds  had  been  made  in  the  mean 
time  of  the  watch  readings  in  the  above  example,  what  error  would 
have  been  produced  in  the  resulting  azimuth  of  the  line  A£J 

45.  LATITUDE  BY  THE  SUN. 

When  the  sun  is  on  the  meridian  it  is  at  its  maximum 
altitude  very  nearly,  and  if  this  be  measured  with  a  sextant 

the  latitude  of  the  place  becomes 
known.  Thus  in  the  figure  let  the 
circle  be  a  meridian  section  of  the 
celestial  sphere,  P  the  pole,  Z  the 
zenith,  Q  the  equator,  5  the  sun,  and 
//'//'the  horizon.  The  arc  SH'  is  the 
altitude  of  the  sun,  SQ  is  its  declina- 
tion, and  ZQ  is  the  latitude  of  the 
place  of  observation  A.  Let  //  be 
the  meridian  altitude,  corrected  for  refraction  and  parallax, 
6  the  declination  of  the  sun,  and  0  the  latitude  of  the  place. 
Then,  from  the  figure, 

0  =  90°  "  h  +  <$,  (45) 

in  which  <$"  is  positive  when  the  sun  is  north  of  the  equator 
and  negative  when  it  is  south  of  the  equator. 

On  the  ocean  the  altitude  is   taken  by  bringing  down  the 


Q 


45»  LATITUDE   BY  THE   SUN.  12$ 

image  of  the  sun  until  its  lower  limb  touches  the  sea  horizon. 
On  land  the  image  is  brought  down  until  the  lower  limb 
touches  its  reflection  as  seen  in  a  dish  of  mercury,  and  thus 
the  double  altitude  is  read.  The  operations  are  begun  several 
minutes  before  apparent  noon  and  a  number  of  measurements 
made  which  give  altitudes  gradually  increasing  to  a  maximum 
and  then  decreasing.  The  proper  corrections  are  then  to  be 
applied  to  the  maximum  altitude,  and  finally  the  above 
formula  gives  the  latitude,  d  being  taken  from  the  nautical 
almanac. 

For  example  on  October  i,  1897,  the  maximum  double 
altitude  of  the  sun's  lower  limb,  observed  with  a  sextant  by 
a  student  at  Lehigh  University  and  corrected  for  index  error 
and  eccentricity  of  the  instrument,  was  91°  51'  if  .2,  one-half 
of  which,  or  45°  55'  38". 6,  is  the  apparent  altitude  of  the 
lower  limb.  To  this  is  to  be  added  16'  oi".6  for  the  sun's 
semi-diameter,  giving  46°  n'  40". 2  apparent  altitude  of  the 
sun's  center.  The  refraction  correction  to  be  subtracted  is 
o'  53//.  I  and  the  parallax  correction  to  be  added  is  06". 2,  and 
thus  the  true  altitude  h  was  45°  $4'  51". 6.  The  declination 
of  the  sun  being  S.  3°  28'  44". 3,  as  found  from  the  nautical 
almanac,  taking  account  of  the  difference  in  longitude,  the 
latitude  of  the  place  by  (45)  is  40°  36'  24",  a  result  whose 
probable  error  is  5"  or-  more,  since  but  a  single  reading  was 
taken  on  the  sextant. 

A  more  precise  determination  can  be  made  by  taking  about 
six  altitudes  at  intervals  of  one  minute,  half  being  before  and 
half  after  the  time  of  maximum  altitude.  It  is  well  also  that 
three  of  them  should  be  taken  by  bringing  the  sun's  lower 
limb  into  coincidence  with  its  image  in  the  mercury,  and 
three  by  bringing  the  upper  limb  to  coincide  with  its  image  j 
the  mean  of  the  six  then  gives  the  double  apparent  altitude 
of  the  sun's  center.  This  altitude,  after  correction  for  the 
errors  of  the  instrument  and  for  parallax  and  refraction,  may 
be  safely  used  to  give  a  latitude  determination  with  a  prob- 

UNIVERSITY  OF  CALIFORNIA 

NT  OF  CIVIL   ENGINEERS 
ECKKEi-EY.  CALIFOi-tMIA 


T26  ASTRONOMICAL   WORK.  V. 

<fcle  error  less  than  5",  and  this  can  be  rendered  smaller  by 
combining  the  results  of  several  observations  made  on  differ- 
ent days.  Some  observers  apply  to  each  of  the  altitudes  a 
torrection  to  reduce  them  to  the  meridian,  but  this  requires 
a  knowledge  of  the  local  time  and  longitude,  and  the  com- 
putation generally  involves  more  labor  than  is  justified  by  the 
precision  of  the  vertical  angles  taken  with  a  sextant  or 
engineers'  transit. 

When  the  local  mean  solar  time  is  known,  or  when  it  can 
be  obtained  from  a  watch  indicating  standard  time,  the 
observation  may  be  made  at  any  hour-angle  with  results  as 
satisfactory  as  those  found  from  noonday  work.  The  alti- 
tudes may  be  taken  at  even  minutes  of  time,  or  the  sextant 
may  be  set  at  even  minutes  of  angle  and  the  watch  be  read 
to  seconds  of  time.  Half  the  altitudes  being  taken  upon  the 
lower  limb  and  half  upon  the  upper,  the  mean  of  all  furnishes 
the  altitude  of  the  sun's  center,  and  the  mean  of  the  recorded 
times  gives  the  corresponding  time  from  which  the  apparent 
solar  time  and  the  hour-angle  t  are  found.  Let  V  be  an  angle 
computed  from 

tan  V  =  tand/cos/, 

then  the  latitude  0  is  found  from 

cos(0  —   V)  —  sin  V  sinh/sind.  (45)' 

For  example,  ten  altitudes  measured  in  about  six  minutes  on 
the  afternoon  of  September  27,  1897,  at  South  Bethlehem, 
Pa.,  gave  after  correction  the  mean  altitude  //  =  36°  03' 
09". 4,  and  the  mean  of  the  ten  times  of  observation  was 
2h  i6m  i88.9  in  local  mean  solar  time.  Applying  the  equation 
of  time  for  that  day,  the  local  apparent  solar  time  is  found  to 
be  2*  25™  34s.  i,  and  hence  the  hour-angle  of  the  sun  is 
/  =  36°  23'  3 1 ".5.  From  the  nautical  almanac,  knowing  that 
the  local  time  of  the  place  of  observation  is  5h  im  32s  slower 
than  Greenwich  time,  the  declination  of  the  sun  is  —  i°  57' 


46.  LATITUDE   BY   A  STAR.  I2/ 

32". 8.  Then  by  computation  V  is  —  2°  25'  59". 6  and  finally 
the  latitude  0  is  found  to  be  +  40°  36'  24". 6. 

The  probable  error  of  a^single  latitude  determination  made 
in  this  manner  is  about  2"  or  3".  To  obtain  more  precise 
results  a  star  should  be  observed,  and  in  general  all  astro- 
nomical work  on  the  sun  has  a  much  lower  degree  of  precision 
than  that  done  on  the  stars. 

Prob.  45.  From  the  values  of  #,  /,  and  ^,  as  stated  above,  com- 
pute the  values  of  Fand  0.  Also  using  the  values  of  #,  h,  and  0, 
compute  t  from  formula  (47). 

46.  LATITUDE  BY  A  STAR. 

The  altitude  of  the  celestial  pole  above  the  horizon  of  any 
place  is  the  latitude  of  that  place.  Hence  when  a  circum- 
polar  star  crosses  the  meridian  its  altitude  plus  or  minus  its 
co-declination  gives  the  latitude  of  the  place,  or 

0  =  h  ±  (90°  --  <*),  (46) 

the  plus  sign  being  used  for  the  lower  culmination  and  the 
minus  sign  for  the  upper  culmination.  As  the  times  of  the 
culmination  of  Polaris  are  given  in  surveyors'  handbooks  this 
method  is  well  adapted  tt>  observations  upon  it  with  the 
sextant  or  with  the  engineers'  transit.  If  the  altitude  kl  be 
observed  at  upper  and  h^  at  lower  culmination,  then  the  mean 
of  these,  each  being  corrected  for  refraction,  gives  the  lati- 
tude, or  0  =  £(/*,  +./*,).  Here  hl  may  be  itself  the  mean  of 
several  altitudes  taken  at  equal  intervals  before  and  after  the 
upper  culmination  and  h^  may  be  the  mean  of  several  similarly 
taken  before  and  after  the  lower  culmination.  With  a  good 
sextant  the  latitude  may  be  found  by  a  few  series  of  observa- 
tions with  a  probable  error  of  one  or  two  seconds  of  angle, 
and  with  a  transit  to  a  less  degree  of  precision. 

Formula  (45)'  of  the  last  Article  may  be  used  to  find  the 
latitude  from  an  observed  altitude  of  any  star  in  any  position, 
if  its  hour-angle  t  is  known.  When  a  sidereal  chronometer  is 


128  ASTROKOiMICAL   WORK.  V. 

at  hand  the  sidereal  time  of  taking  the  altitude,  diminished 
by  the  right  ascension  of  the  star,  gives  the  hour-angle  in 
sidereal  time,  and  fifteen  times  this  is  the  value  of  t  in 
angular  measure.  For  example  on  December  8,  1897,  ten 
altitudes  of  Polaris  were  taken  with  a  sextant  in  about  thirteen 
minutes  at  Lehigh  University,  the  time  of  each  being  noted 
on  a  sidereal  chronometer.  The  mean  apparent  altitude,  after 
correction  for  index  error  and  eccentricity,  was  41°  47'  46". 3, 
and  applying  the  refraction  correction  i'  06". 3,  the  mean  true 
altitude  is  //  —  41°  46'  40". o.  The  mean  sidereal  time  of 
the  times  of  the  ten  measurements  was  oh  iom  54s. 9,  from 
which  is  subtracted  the  right  ascension  of  Polaris  or  ih  50'" 
33s. 6  to  give  its  hour-angle  22h  50™  33S.6,  whose  equivalent  in 
degree  measure  is  /  =  342°  38'  24''. o.  The  declination  of 
Polaris  being  #  =  88°  46'  n".gt  the  auxiliary  V  is  found  to 
be  88°  49'  33". 5,  and  then  from  (45)'  there  results  for  the 
latitude  the  value  0  =  40°  $6'  17".$.  The  probable  error  of 
a  single  determination  of  latitude  made  in  this  manner  is 
much  less  than  that  of  one  found  from  observations  on  the 
sun,  say  about  i"  or  2" .  When  a  common  watch  is  used  its 
error  and  rate  should  be  known  so  that  the  time  correspond- 
ing to  the  mean  altitude  may  be  converted  into  local  mean 
solar  time  and  then,  with  the  help  of  the  nautical  almanac, 
into  sidereal  time,  from  which  the  hour-angle  t  is  found  as 
before.  It  is  preferable  that  the  student  should  use  a  com- 
mon watch  rather  than  a  sidereal  chronometer,  since  the 
former  is  more  generally  at  hand  in  actual  work. 

The  best  time  for  making  this  observation  is  when  the  star 
is  near  culmination,  since  then  an  error  in  h  produces  the 
smallest  error  in  0.  In  the  above  example  the  star  was 
about  ih  iom  from  the  lower  culmination  and  hence  in  a 
favorable  position. 

Prob.  46.  Deduce  sin  h  =  sin  0  sin  d  +  cos  0  cos  6  cos  /,  and 
show  that  an  error  dh  gives  the  least  error  d(t>  when  /  =  o°  or  when 
/=  1 80°. 


47-  TIME.  129 

*  47.  TIME. 

A  watch  may  he  set  to"focal  apparent  solar  time  by  noting 
the  instant  when  the  sun  attains  its  maximum  altitude,  and 
then,  applying  the  equation  of  time,  local  mean  solar  time  is 
approximately  known.  At  any  telegraph  station  in  the 
United  States  a  watch  may  be  closely  set  to  standard  time, 
Eastern  standard  time  being  the  mean  solar  time  of  the  75th 
meridian  and  hence  five  hours  slower  than  Greenwich  mean 
solar  time,  while  Central,  Mountain,  and  Pacific  standard 
times  are  the  mean  solar  times  of  the  QOth,  iO5th,  and  i2Oth 
meridians  respectively.  The  mean  solar  time  at  any  other 
meridian  is  found  from  standard  time  by  applying  to  standard 
time  the  correction  for  difference  of  longitude,  15  degrees 
corresponding  to  one  hour  of  time.  When  greater  precision 
is  required  an  altitude  of  the  sun  or  of  a  star  is  to  be  taken 
and  from  this  the  error  of  the  watch  can  be  computed,  if  the 
latitude  of  the  place  is  known. 

In  the  figure  of  Art.  42  let  5  be  the  place  of  the  sun  or 
star;  the  arc  SZ  is  the  co-altitude  or  zenith  distance  £,  the 
arc  SP  is  the  co-declination  90°  -  d,  the  arc  ZP  is  the 
co-latitude  90°  —  0,  and  the  angle  SPZ  is  the  hour-angle  of 
the  star,  which  is  designated  by  /.  The  solution  of  the 
spherical  triangle  gives  i 

cos-sr  =  sin0  sintf  -f-  cos0  costf  cos/, 

from  which  /  can  be  computed;  it  is,  however,  customary  to 
reduce  this  equation  to  the  form 

tan*/  =  A  /sin«*  +  0  -  V    sin*<*  ~  *+~^        (47) 
V  cosi(*  +  0  +  d)    cosi(*  —  0  -  6) 

From  this  /  is  found  in  degrees,  minutes,  and  seconds,  and 
this  value  is  then  changed  into  time  by  dividing  it  by  15. 
When  the  sun  is  observed  this  result  is  apparent  solar  time; 
when  a  star  is  observed  its  sidereal  time  interval  is  to  be 
reduced  to  mean  solar  time. 


130  ASTRONOMICAL   WORK.  V. 

For  example,  take  the  data  of  Art.  42  where  observations 
on  the  sun  gave  the  mean  corrected  co-altitude  z  =  46°  02' 
32"  at  8h  43ra  oos  A.M.  by  the  watch,  the  sun's  declination 
being  d  —  19°  54'  05"  and  the  latitude  of  the  place  0  =  40° 
36'  27".  Inserting  these  values  in  the  above  formula  there  is 
found  t  =  —  48°  32'  50",  which  is  the  hour-angle  between  the 
sun  and  the  meridian;  this  reduced  to  time  gives  3h  14™  ns.3 
as  the  interval  between  the  time  of  the  mean  altitude  and  that 
of  apparent  solar  noon.  Hence  8h  45m  588.7  was  the  local 
apparent  solar  time,  and  subtracting  the  equation  of  time  for 
the  given  day,  there  results  8h  42™  I58.5  as  the  local  mean 
solar  time  which  corresponded  to  8h  43™  oo8  of  the  watch. 
Hence  the  deviation  of  the  watch  from  local  mean  solar  time 
was  om  44s. 5  fast.  Further  as  the  place  of  observation  was 
o°  22'  35"  west  of  the  75th  meridian,  and  as  this  corresponds 
to  oh  oim  3O8.3,  it  follows  that  the  deviation  of  the  watch  from 
Eastern  standard  time  was  45s. 8  slow.  The  probable  error  of 
this  determination  may  be  several  seconds. 

Far  better  work  can  be  done  by  observing  a  star,  and  a 
good  sextant  is  always  to  be  preferred  to  an  engineer's  transit 
for  taking  the  altitudes,  the  image  being  brought  down  to 
coincide  with  its  reflection  in  a  dish  of  mercury.  The  follow- 
ing is  a  record  of  an  observation  made  at  Lehigh  University 
on  May  9,  1899,  by  this  method.  The  watch  was  supposed 
to  carry  Eastern  standard  time  and  it  was  required  to  deter- 
mine its  error.  The  sextant  was  set  successively  at  even  10 
seconds  of  arc  and  the  watch  time  of  each  noted;  thus  the 
observed  mean  double  altitude  90°  05'  oo"  occurred  at 
8h  02m  368.64  by  the  watch.  This  is  corrected  for  index 
error  and  eccentricity,  and  the  apparent  double  altitude 
found,  to  which  a  refraction  correction,  computed  from  a 
formula  that  takes  barometer  and  thermometer  into  account, 
is  applied.  Thus  the  true  zenith  distance  z  is  found,  and 
from  this  and  the  given  latitude  of  the  place  and  declination 
of  the  star  the  hour-angle  t  is  computed  from  the  above 


47- 


TIME. 
TIME   BY  a  GEMINORUM  (CASTOR),  MAY  9,  1899. 


Obs. 

No. 

Double 
Altitude. 

Watch  Time 

P.M.        **' 

Data  and  Remarks. 

I 

90°  50' 

81'  oom  36*.  5 

J.  H.  O.,  observer.     C.  L.  T.,  recorder. 

2 

90    40 

,   oi    03  .5 

Pistor  and   Martin's  Prismatic   Sextant 

3 

90    30 

oi    30  .5 

with  mercury  horizon  and  glass  cover. 

4 

90    20 

oi    56  .5 

Watch    carrying    approximate   Eastern 
standard  time. 

5 
6* 

90   10 
90  oo 

02      23  .5 
O2      5O  .2 

0  =  4o'36'23".2                  Index  Error, 
from  Arcturus. 

A  =  75°  22'  57".3                             ,       „ 

7 

89  50 

03    16  .6 

—  10    45 

a  =    7h  28™  ii».  30                           , 

8 

89   40 

03    42  .5 

d  -  32°  06'  36".  7 

9 

89  30 

04    ii  .4 

—  I0     55 
Barometer,  29'°  .550 

10 

89    20 

04    35  -8 

—  io     55 

Aftir'Tirrl   tTi  C-T  m      f\f\Q   r\  P 

Detached  therm.,  64.7  F. 

Means 

90°  05'  oo" 

8h  02m  36s.  64 

*  Horizon  cover  reversed. 

Observed  zh  =  90°  05'  oo"        True  sidereal  time  /-(-  a=  iih  i2m  io8.63 

Index  error  =      —  16    51.25     Sidereal  time  mean  noon  =    3    09    09  .27 

Eccentricity  -             -  48.85     Sidereal  intervai  after 

2k  =  89    47    25.90         mean  n°°n                        =    8   03    oi  .36 

Apparent  ^=44    53    42.95     Correction  to  mean  solar  ^    _  ^     ^    ^ 

J^gfj""*  **+  *  '"**•»    —                          *  f    *£* 

33-t 

Local  mean  solar  time  =     8   oi    42  .07 

True/&=44    52    47-49    Reduction  to  75th  meridian  =   +  oi    31.82 

Hour-angle/-   55°  59'   50"             Eastern  standard  tims  =     803     13.89 

Hour/=     3h43M59'.33                              Watch  time  =     802    36.64 

Watch  error  (slow)  =                  37*-25 

C.  L.  T.,  computer. 

formula,  and  converted  into  time  measure.  The  true  sidereal 
time  then  results  by  adding  the  right  ascension  of  the  star, 
and  this  is  converted  into  local  mean  solar  time  and  then  into 
Eastern  standard  time,  from  which  finally  the  watch  error  is 
found  to  have  been  37.25  seconds  slow.  The  probable  error 
of  this  determination  is  less  than  one-quarter  of  a  second. 
The  most  favorable  position  of  a  star  for  this  work  is  when 


132  ASTRONOMICAL   WORK.  V. 

it  is  on  the  prime  vertical.  For,  if  dz  be  an  error  in  2,  the 
corresponding  error  dt  in  /,  obtained  by  differentiating  the 
first  equation  of  this  Article,  is  sin^-^/cos^  costf  sin/,  and 
since  sin^/costf  sin/  =  sinzT,  this  gives  dt  =  dz/cvsty  s'mZ. 
Accordingly  the  azimuth  Z  should  be  90°  or  270°  in  order 
that  dt  may  have  its  smallest  value.  In  the  same  manner  it 
is  shown  that  errors  in  the  assumed  latitude  produce  the  least 
effect  when  the  star  is  on  the  prime  vertical. 

Prob.  47.  Using  the  data  of  the  above  observation  on  Castor,  find 
the  error  in  the  computed  Eastern  standard  time  which  would  be 
caused  by  an  error  of  01"  in  the  altitude  h  ;  also  the  error  which 
would  be  caused  by  an  error  of  01"  in  the  latitude  0. 

48.  LONGITUDE. 

When  accurate  standard  time  is  at  hand  the  comparison  of 
It  with  the  local  mean  time  gives  the  longitude.  Thus,  if  the 
local  mean  solar  time  of  a  place  has  been  found  by  a  star 
observation  to  be  I4mo8s.4  faster  than  Central  standard  time, 
the  place  is  3°  22'  04"  east  of  the  90°  meridian  and  hence  its 
longitude  is  86°  27'  $6"  west  of  Greenwich.  This  method  is 
used  at  sea,  where  daily  observations  for  local  mean  solar 
time  are  made  on  the  sun  or  stars  when  the  weather  permits, 
this  local  time  being  compared  with  a  chronometer  which 
indicates  either  Greenwich  mean  solar  time  or  that  of  a  port 
whose  longitude  is  known.  As  one  second  of  time  is  equiva- 
lent to  fifteen  seconds  of  angle,  it  is  seen  that  this  method  is 
not  very  precise,  particularly  when  it  is  considered  that  the 
best  watches  are  liable  to  vary  one  or  more  seconds  per  day. 

The  method  of  lunar  distances  is  extensively  used  at  sea 
for  finding  the  Greenwich  time.  In  the  nautical  almanac  will 
be  found  the  true  angular  distances  between  the  moon's 
center  and  several  stars  and  planets  for  every  day  in  the  year 
and  for  three-hour  intervals,  these  distances  being  stated  as 
they  would  appear  from  the  center  of  the  earth.  If  one  of 
these  apparent  distances  be  measured  at  any  place,  as  also 


48.  LONGITUDE.  133 

the  apparent  altitudes  of  the  star  and  moon,  the  data  are  at 
hand  for  computing  the  true  distance  as  seen  from  the  center 
of  the  earth  at  the  same  instant,  and  thus  from  the  almanac 
Greenwich  mean  time  is  known.  Then,  the  difference 
between  local  and  Greenwich  time  gives  the  longitude  of  the 
place.  This  method  involves  laborious  computations  unless 
special  tables  are  at  hand. 

Anothec  method  is  that  of  lunar  culminations  which  re- 
quires that  azimuth  and  time  should  have  been  determined. 
The  instant  of  the  passage  of  the  moon's  bright  limb  across 
the  meridian  is  observed,  and  a  correction  applied  to  find  the 
local  mean  time  of  passage  of  the  moon's  center.  This  local 
mean  time,  converted  into  sidereal  time,  furnishes  the  right 
ascension  of  the  moon,  while  the  Greenwich  mean  time 
corresponding  to  the  same  right  ascension  can  be  found  from 
the  almanac.  Lastly  the  difference  between  local  and  Green- 
wich mean  time  gives  the  longitude  of  the  place. 

As  an  example  of  the  method  of  lunar  culminations  the 
following  rough  observation  with  an  engineer's  transit,  made 
at  Lehigh  University  on  May  23,  1899,  may  be  of  interest. 
On  that  day  the  moon  crossed  the  meridian  at  about  ioh  55™ 
P.M.,  and  it  was  accordingly  arranged  to  determine  azimuth 
by  pointing  on  Polaris  a  few  minutes  previous.  By  a  simple 
computation  it  was  determined  that  the  azimuth  of  Polaris  at 
ioh  45m  local  mean  time  was  180°  36'  23". 4  and  at  that 
instant  the  cross-hair  of  the  transit  telescope  was  set  on  the 
star.  Then  the  angle  o°  36'  20"  was  turned  off  toward  the 
west  and  the  telescope  reversed,  thus  pointing  southward  in 
the  plane  of  the  meridian.  When  the  moon's  west  limb 
touched  the  vertical  cross-hair  the  time  was  noted  as  ioh  53m 
34s.  i.  Reducing  this  to  sidereal  time  with  the  assumed 
longitude  5h,  and  adding  a  correction  for  the  time  required 
for  the  semi-diameter  of  the  moon  to  pass  the  meridian,  the 
right  ascension  of  the  moon's  center  when  crossing  the 
meridian  of  the  place  is  found  to  be  I5h  oom  538.44,  and  the 


134  ASTRONOMICAL   WORK.  V. 

corresponding  mean  local  time  i~oh  54'°  558.59.  From  the 
nautical  almanac  the  Greenwich  mean  time  at  which  the 
moon's  center  had  this  right  ascension  is  found  to  be  I5h  54™ 
27s. 84,  and  consequently  the  longiti^de  of  the  place  of  obser- 
vation is  4h  59m  428.25  in  time  or  74°  55'  34"  in  arc,  a  result 
which  is  in  error  by  nearly  half  a  degree,  the  true  value  being 
75°  22'  23". 

It  thus  appears  that  no  close  determination  of  the  longitude 
of  a  place  can  be  made  by  the  method  of  moon  culminations 
with  an  engineer's  transit.  Nevertheless  in  an  unexplored 
region  the  method  is  of  value  in  making  an  approximate 
determination  to  be  used  in  time  observations  and  in  taking 
quantities  from  the  nautical  almanac. 

Prob.  48.  At  a  certain  place  on  December  5,  1900,  the  moon's 
right  ascension  was  observed  as  4h  32™  O5s.3i  at  8  P.M.  local  mean 
time.  From  the  nautical  almanac  it  is  found  that  the  right  ascen- 
sions 4h  31™  039.69  and  4h  33™  37S.82  occurred  at  3  A.M.  and  4  A.M., 
Greenwich  mean  time,  on  December  6,  1900.  Find  the  longitude 
of  the  place  of  observation. 

49.  PRECISE  DETERMINATIONS. 

The  methods  set  forth  in  the  preceding  pages  give  results 
whose  precision  is  far  lower  than  that  needed  for  the  astro- 
nomical work  of  a  geodetic  survey.  When  it  is  required  to 
determine  azimuth,  latitude,  and  longitude  at  one  of  the 
stations  of  a  geodetic  triangulation  such  methods  are  generally 
used  to  furnish  preliminary  approximate  values,  for  it  has 
been  seen  that  each  of  these  elements  depends  upon  the 
others,  and  hence  rough  methods  must  precede  precise  ones. 
These  preliminary  values  may  be  supposed  to  give  the  lati- 
tude within  one  or  two  seconds,  the  longitude  within  ten  or 
twenty  seconds,  and  the  azimuth  within  six  or  eight  seconds. 

To  make  a  precise  determination  of  azimuth  a  direction 
theodolite,  having  a  circle  divided  to  5  minutes  and  reading 
by  microscopes  to  seconds  or  less,  is  used.  The  observations 


49-  PRECISE   DETERMINATIONS.  135 

are  made  on  close  circumpolar  stars  by  the  method  of  Art. 
35,  great  pains  being  taken  to  eliminate  the  error  of  level  in 
the  horizontal  axis  of  the  telescope.  By  making  a  sufficient 
number  of  measurements  the  azimuth  of  a  line  running  from 
the  station  to  a  signal  may  be  found  with  a  probable  error  of 
i"  or  less,  and  by  measuring  the  angle  between  this  line  and 
one  of  the  sides  of  the  triangulation  the  azimuth  of  the  latter 
is  known  with  almost  equal  precision. 

To  make  a  precise  determination  of  latitude  a  zenith  tele- 
scope is  to  be  set  up  in  the  plane  of  the  meridian  and  the 
difference  of  the  meridian  zenith  distances  of  two  stars  that 
cross  the  meridian  near  the  zenith,  but  on  opposite  sides  of 
it,  is  observed.  Let  #,  and  <?,  be  the  declinations  of  the  two 
stars,  the  first  being  south  of  the  zenith,  zl  and  z^  their 
apparent  zenith  distances,  and  r,  and  r^  their  refraction  cor- 
rections. Then  for  the  first  star  0  =  #,  -\-  zl  -\-  r^  and  for 
the  other  0  ==  d,  —  #a  —  r,.  The  addition  of  these  gives 

20  =  *,  +  S,  +  (,t  -  z,)  +  (r,  -  r,), 

so  that  it  is  only  necessary  to  measure  st  —  zt  by  the  mi- 
crometer in  the  field  of  the  zenith  telescope  and  then  apply 
the  small  refraction  correction.  By  this  method  it  is  easy 
to  determine  latitude  with  a  probable  error  less  than  o" .  I. 

To  make  a  precise  determination  of  longitude  a  telegraph 
line  must  connect  the  station  with  an  observatory  whose 
longitude  is  known.  A  portable  astronomical  transit  instru- 
ment is  mounted  in  the  plane  of  the  meridian  and  the  time 
of  passage  of  several  equatorial  stars  is  signalled  by  the  tele- 
graph line  to  the  observatory.  When  the  same  stars  pass  the 
meridian  of  the  observatory  their  time  of  passage  is  signalled 
to  the  station.  A  single  clock  in  the  telegraph  circuit  may 
be  used  to  make  a  chronographic  record  of  both  series  of 
signals,  and  thus  the  difference  in  time  is  known,  from  which 
the  longitude  directly  results.  The  probable  error  of  the 
difference  of  longitude  thus  determined  may  be  made  as 
small  as  o8  .01  or  c/'.is. 


136  •  ASTRONOMICAL   WORK.  V. 

Numerous  observations  made  at  many  different  observa- 
tories have  established  the  fact  that  small  periodic  changes  in 
the  latitudes  of  all  places  are  constantly  going  on.  This  is 
due  to  a  slight  wabbling  motion  of  the  earth's  mass  with 
respect  to  its  axis,  so  that  the  axis  performs  an  apparent 
revolution  around  its  mean  position  in  about  425  days,  and 
consequently  the  north  pole  of  the  earth  makes  a  similar 
revolution  around  its  mean  position.  The  radius  of  this 
circle  varies  from  o".  16  to  o".36,  and  consequently  the  lati- 
tude of  any  given  point  on  the  earth's  surface  may  vary  from 
o".32  to  o".72  at  different  times.  It  is  hence  seen  that 
decimals  of  seconds  occurring  in  common  latitude  determina- 
tions have  no  definite  meaning.  As  all  methods  for  determin- 
ing the  azimuth  of  a  line  involve  a  knowledge  of  the  latitude 
of  that  end  where  the  observation  is  made,  it  follows  that  the 
astronomical  azimuths  of  all  lines  on  the  earth's  surface  also 
undergo  periodic  changes;  and  the  same  holds  true  for  longi- 
tudes of  places.  These  changes  will  be  the  greater  the 
nearer  the  line  or  place  is  to  the  north  pole,  but  near  the 
equator  they  will  be  very  small. 

In  the  science  of  geodesy  the  words  azimuth,  latitude,  and 
longitude  have  a  signification  slightly  different  from  that  in 
astronomy,  as  will  be  seen  in  Chapter  VII.  These  geodetic 
elements  enable  a  fixed  system  of  coordinates  to  be  estab- 
lished by  which  the  relative  positions  of  points  on  the  earth's 
surface  can  be  expressed  to  a  degree  of  precision  limited  only 
by  our  knowledge  of  the  shape  and  size  of  the  earth. 

Prob.  49.  Consult  Albrecht's  Bericht  tiber  den  Stand  der  Erfor- 
schung  der  Breitenvariation  (Berlin,  1899),  and  give  a  sketch  show- 
ing how  the  true  north  pole  moves  around  its  mean  position. 


£0.  EARLY   HISTORY.  137 


ft 
4 


CHAPTER  VI. 
SPHERICAL    GEODESY. 

50.  EARLY  HISTORY. 

Geodesy  is  the  science  that  sets  forth  the  principles  and 
methods  whereby  large  areas  on  the  surface  of  the  earth  may 
be  surveyed  and  mapped  with  precision.  If  the  surface  of 
the  earth  were  a  plane,  as  certain  ancient  peoples  supposed, 
the  science  of  geodesy  could  never  have  arisen,  since  the 
elementary  geometry  of  Euclid  would  be  capable  of  measur- 
ing and  representing  its  geographical  features.  In  fact, 
however,  measurements  conducted  upon  this  supposition 
become  more  or  less  entangled  in  discrepancies  according  to 
the  size  of  the  country  over  which  they  are  carried.  For 
instance,  let  three  points  be  taken  on  the  earth's  surface  at 
considerable  distances  apart;  the  sum  of  the  three  angles  of 
the  triangle  thus  formed  is  found,  if  measured  by  an  instru- 
ment whose  graduated  arc  is  placed  level  at  each  station,  to 
be  greater  than  180  degrees.  From  these  and  many  other 
discrepancies  it  is  to  be  concluded  that  the  earth's  surface  is 
not  a  plane. 

Many  facts  are  known  from  which  it  is  inferred  that  the 
earth  is  globular,  such  as  the  appearance  of  the  top  of  a 
light-house  earlier  than  its  base  to  a  ship  approaching  the 
shore,  the  dip  of  the  sea  horizon,  the  elevation  of  the  pole 
star  as  we  travel  north  and  its  depression  as  we  travel  south, 
the  analogy  of  the  other  planets  which  seem  to  be  globular 
when  viewed  through  a  glass;  and  the  circular  form  of  the 


138  SPHERICAL   GEODESY.  VI. 

earth's  shadow  as  observed  in  a  lunar  eclipse.  To  these 
must  be  added  the  well-known  circumstance  that  travellers, 
going  ever  eastward,  pass  entirely  around  the  earth  and  return 
to  the  point  of  starting.  From  these  facts  ib  is  concluded 
that  the  earth  js  globular,  that  is  to  say  like  a  globe,  but 
whether  spherical,  spheroidal,  or  ellipsoidal,  there  is  thus  far 
no  evidence. 

The  surface  whose  size  and  shape  is  to  be  investigated  in 
the  following  pages  is  that  of  the  great  ocean  which  covers 
fully  Hiree- fourths  of  the  globe.  Although  this  is  agitated 
by  winds  and  raised  in  tides  its  mean  level  can  be  accurately 
determined.  Moreover  the  land  is  really  elevated  but  little 
above  the  ocean,  for  it  is  now  known  that  the  radius  of  the 
earth,  regarded  as  a  sphere,  is  nearly  4000  miles,  while  the 
highest  mountains  rise  only  about  5  miles.  Hence  measure- 
ments made  upon  the  land  can  at  the  utmost  cause  an  error 
of  only  one  eight-hundredth  part  in  the  value  of  the  radius. 

The  early  Greek  philosophers  speculated  upon  the  shape 
of  the  earth.  Anaximander  (570  B.C.)  called  it  a  cylinder 
whose  height  was  three  times  its  diameter,  the  land  and  sea 
being  on  its  upper  base.  Plato  (400  B.C.)  thought  it  a  cube. 
Aristotle  (340  B.C.)  gives  reasons  for  supposing  it  to  be  a 
sphere  and  mentions,  as  also  does  Archimedes  (250  B.C.), 
that  geometers  had  estimated  its  circumference  at  300  ooo 
stadia.  The  first  recorded  observations  for  determining  the 
size  of  the  sphere  are,  however,  those  made  in  Egypt  by 
Eratosthenes  (230  B.C.);  his  method,  though  rude  in  meas- 
urement, is  correct  in  principle,  and  from  it  he  concluded  that 
the  circumference  of  the  earth  was  250000  stadia. 

The  process  by  which  Eratosthenes  deduced  the  size  of  the 
earth  will  now  be  described.  He  noticed  that  at  Syene  in 
southern  Egypt  the  sun  on  the  day  of  the  summer  solstice 
cast  no  shadow  of  a  vertical  object,  while  at  Alexandria  in 
northern  Egypt  the  rays  of  the  sun  on  the  same  day  of  the 


50.  EARLY   HISTORY.  139 

year  made  an  angle  with  the  vertical  of  one-fiftieth  of  four 
right  angles.  From  this  he  concluded  that  the  distance 
between  Syene  and  Alexandria  was  one- 
fiftieth  of  the  circumference  of  the  earth, 
and  as  that  distance  was  about  5  ooo  stadia 
he  claimed  the  whole  circumference  to  be 
250000  stadia.  The  exact  length  of  the 
stadium  is  now  unknown,  so  that  the  pre- 
cision of  his  result  cannot  be  judged,  yet 
the  name  of  Eratosthenes  will  ever  be 
honored  in  science  as  the  originator  of  the 
method  of  deducing  the  size  of  the  earth 
from  a  measured  meridian  arc. 

To  explain  the  reasoning  of  Eratosthenes  let  the  figure 
represent  a  meridian  section  of  the  earth,  PP'  being  the  axis, 
QQ  the  equator,  P  the  north  pole,  A  the  position  of 
Alexandria,  and  B  that  of  Syene, -while  AS  and  BS  give  the 
directions  of  the  sun  at  the  summer  solstice.  Assuming  that 
the  section  is  a  circle  and  that  the  rays  of  the  sun  are  parallel 
it  is  clear  that  the  angle  BOA  is  equal  to  the  angle  which  the 
sun's  rays  make  with  OA.  Thus,  if  this  angle  be  -^h  °^  3^° 
degrees,  it  follows  that  the  circumference  is  fifty  times  the 
distance  AB.  The  reasoning  of  Eratosthenes  hence  involves 
two  fundamental  conceptions  besides  those  of  geometry, 
namely,  that  the  earth  is  a  sphere  and  that  the  sun  is  at  a 
great  distance  from  it. 

The  method  of  Eratosthenes  is  called  the  measurement  of 
a  meridian  arc.  Thus,  let  the  distance  between  the  two 
points  A  and  B  be  /,  and  let  the  angle  AOB  be  6  degrees; 
then  in  modern  reasoning, 

1/0  =  length  of  one  degree  of  the  meridian, 
36o//#  =  length  of  circumferences  of  the  earth, 
57.2958/76*  —  length  of  radius  of  the  earth. 

Eratosthenes   found   the  distance  /  from   the    statements  of 


140  SPHERICAL   GEODESY.  VI. 

travellers,  later  observers  rolled  a  wheel,  or  measured  it  with 
a  chain,  but  the  modern  method  is  to  compute  it  from  a  pre- 
cise triangulation.  Eratosthenes  found  the  angle  6  from  the 
shadows  cast  by  vertical  posts,  but  later  observers  found  it 
from  the  latitudes  of  the  places;  for  the  angle  QOA  is  the 
latitude  of  A,  while  Q.OB  is  the  latitude  of  J5,  and  hence  6  is 
the  difference  in  latitude  between  the  two  ends  of  the 
meridian  arc. 

For  several  hundred  years  after  the  time  of  Eratosthenes 
the  doctrine  of  the  spherical  form  of  the  earth  was  generally 
accepted  by  astronomers.  Posidonias  (90  B.C.)  measured  the 
meridian  arc  between  Alexandria  and  Rhodes,  using  a  star  to 
determine  the  latitudes  and  deduced  240  ooo  stadia  for  the 
circumference.  But  this  knowledge  of  the  Greeks  was  all 
lost  as  their  civilization  declined,  and  for  more  than  a 
thousand  years  Europe,  sunk  in  intellectual  darkness,  made 
no  inquiry  concerning  the  size  or  shape  of  the  earth.  Only 
in  Arabia  were  the  sciences  at  all  cultivated  during  this 
period.  There  the  Caliph  Almamoun  summoned  astronomers 
to  Bagdad,  and  one  of  their  labors  was  the  measurement,  on 
the  plains  of  Mesopotamia,  of  an  arc  of  a  meridian  by 
wooden  rods,  from  which  they  deduced  the  length  of  a 
degree  to  be  56!  Arabian  miles,  or  probably  about  71  English 
miles. 

Prob.  50.  If  the  earth  is  represented  by  a  sphere  16  inches  in 
diameter,  what  is  the  height  in  inches  of  the  tallest  mountain  ? 

51.  HISTORY  FROM  1300  TO  1750. 

In  the  year  1322  a  traveller  named  John  Mandeville  wrote 
a  volume  describing  his  journeys;  this  is  generally  regarded 
as  the  earliest  English  prose  work.  In  it  is  a  lengthy  and 
labored  argument  to  prove  that  the  "  lond  and  the  see  ben  of 
rownde  schapp  and  forme  "  and  that  the  circumference  of  the 
earth  has  360  degrees  like  that  of  the  heavens.  He  con- 


SI.  HISTORY   FROM    1 300  TO    1/5 1.  141 

eludes,  "  be  the  Earthe  devysed  in  als  many  parties,  as  the 
Firmament ;  and  lat  every  partye  answere  to  a  Degree  of  the 
Firmament;  and  wytethe  &  wel,  that  aftre  the  auctoures  of 
Astronomye,  700  Furlonges  of  Earthe  answeren  to  a  Degree 
of  the  Firmament;  and  tho  ben  87  Myles  and  4  Furlonges. 
Now  be  that  here  >multiplyed  by  360  sithes;  and  than  thei 
ben  31  500  Myles,  every  of  8  Furlonges,  aftre  Myles  of  oure 
Contree.  So  moche  hathe  the  Earthe  in  roundnesse,  and 
of  heighte  enviroun,  aftre  myn  opynyoun  and  myn  undir- 
stondynge." 

These  views  of  Mandeville  appear  to  have  produced  but 
little  influence,  for  it  was  not  until  the  fifteenth  century, 
when  the  first  gleams  of  light  broke  in  upon  the  darkness  of 
the  middle  ages,  that  men  began  to  think  again  about  the 
shape  and  size  of  the  earth.  Navigators  began  to  doubt  that 
its  surface  was  a  level  plane,  and  here  and  there  one,  like 
Columbus,  asserted  it  to  be  globular.  In  the  sixteenth  cen- 
tury, the  doctrine  of  the  spherical  form  of  the  earth  was  again 
generally  accepted,  and  one  of  the  ships  of  Magellan,  after  a 
three  years'  voyage,  accomplished  its  circumnavigation. 
With  the  acceptance  of  this  idea  arose  also  the  question  as 
to  the  size  of  the  globe,  and  Fernel,  in  1525,  made  a  meas- 
urement of  an  arc  of  a  meridian  by  rolling  a  wheel  from  Paris 
to  Amiens  to  find  the  distance  and  by  observing  the  latitudes 
with  large  wooden  triangles,  from  which  he  deduced  about 
57050  toises  for  the  length  of  one  degree.  At  this  time 
methods  of  precision  in  surveying  were  entirely  unknown. 
In  1617  Snellius  conceived  the  idea  of  triangulating  from  a 
known  base  line,  and  thus,  near  Leyden,  he  measured  a 
meridian  arc  which  gives  55020  toises  for  the  length  of  a 
degree.  Norwood,  in  1633,  chained  the  distance  from 
London  to  York,  and  deduced  57424  toises  for  a  degree. 
Picard,  who  was  the  first  to  use  spider  lines  in  a  telescope, 
remeasured,  in  1669,  the  arc  from  Paris  to  Amiens,  using  a 
base  line  and  triangulation,  and  found  one  degree  to  be 


142  SPHERICAL   GEODESY.  VI. 

57  060  toises.  This  was  the  result  that  Newton  used  when 
making  his  famous  calculation  which  proved  that  the  moon 
gravitated  toward  the  earth. 

The  toise,  it  should  here  be  noted,  was  an  old  French 
measure,  approximately  equal  to  6.3946  English  feet  or  1.949 
meters.  It  is  of  classic  interest  on  account  of  its  use  in  all 
the  early  meridian  arcs  and  in  the  surveys  for  deciding  upon 
the  length  of  the  meter. 

From  1690  to  1718  Cassini  carried  on  surveys  in  France, 
more  precise  probably  than  any  preceding  ones,  and  in  1720 
'he  published  the  following  results  regarding  three  meridian 
arcs: 

Arc.  Mean  Latitude.  Toises  in  One  Degree. 

1  49°  56'  56  970 

2  49    22  57  060 

3  47    55  57098 

From  these  it  appeared  that  the  length  of  a  degree  of  latitude 
increased  toward  the  equator,  or  that  the  earth  was  flatter  at 
the  equator  than  at  the  poles.  In  other  words  he  claimed 
that  the  earth  was  not  spherical  but  spheroidal,  and  that  the 
spheroid  was  a  prolate  one.  From  the  time  men  had  ceased 
to  believe  in  the  flatness  of  the  earth,  and  had  begun  to 
regard  it  as  a  sphere,  their  investigations  had  been  directed 
toward  its  size  alone;  now,  however,  the  inquiry  assumed  a 
new  phase,  and  its  shape  came  up  again  for  discussion. 

A  prolate  spheroid  is  generated  by  an  ellipse  revolving 
about  its  major  axis,  and  an  oblate  spheroid  by  an  ellipse 
revolving  about  its  minor  axis.  The  first  diagram  of  the 
figure  represents  a  meridian  section  of  the  earth  regarded  as 
a  prolate  spheroid,  and  the  second  shows  the  section  of  an 
oblate  spheroid.  In  each  diagram  PP  is  the  axis,  Q.Q  the 
equator,  and  A  a  place  of  observation  whose  horizon  is  AH> 
zenith  Z,  latitude  ABQ,  and  radius  of  curvature  AR.  Now 
if  the  earth  be  regarded  as  a  sphere  and  its  radius  be  found 
from  a  meridian  arc  near  A,  the  value  AR  will  result.  In 


HISTORY    FROM    1300  TO    1751, 


H3 


the  prolate  spheroid  the  radius  of  curvature  is  least  at  the 
poles  and  greatest  at  the  equator,  and  the  reverse  in  the 
oblate.  Hence  if  the  lengths  of  the  degrees  of  latitude 
decrease  from  the  equator  to  the  poles,  it  shows  that  the 


earth  is  prolate;  but  if  they  increase  from  the  equator  toward 
the  poles,  it  is  a  proof  that  it  is  oblate  in  shape. 

It  is  now  necessary  to  go  back  to  the  year  1687,  the  date 
of  the  publication  of  the  first  edition  of  Newton's  Principia. 
In  Book  III  of  that  great  work  are  discussed  the  observations 
of  Richer,  who,  having  been  sent  to  Cayenne,  in  equatorial 
South  America,  on  an  astronomical  expedition,  noted  that 
his  clock,  which  kept  accurate  time  in  Paris,  there  continually 
lost  two  seconds  daily,  and  could  only  be  corrected  by 
shortening  the  pendulum.  Now,  the  time  of  oscillation  o£  a 
pendulum  of  constant  length  depends  upon  the  intensity  of 
the  force  of  gravity,  and  Newton  showed,  after  making  due 
allowance  for  the  effect  of  centrifugal  force,  that  the  force  of 
gravity  at  Cayenne,  compared  with  that  at  Paris,  was  too 
small  for  the  hypothesis  of  a  spherical  globe;  in  short,  that 
Cayenne  was  further  from  the  center  of  the  globe  than  Paris, 
or  that  the  earth  was  an  oblate  spheroid  flattened  at  the 
poles.  He  computed,  too,  that  the  amount  of  this  flattening 
at  both  poles  was  between  TJT  and  -gfa  of  the  whole  diameter. 
Now  it  will  be  remembered  that  Newton's  philosophy  did 


144  SPHERICAL   GEODESY.  VI. 

not  gain  ready  acceptance  in  France;  this  investigation,  in 
particular,  called  forth  much  argument,  and  when  Cassini's 
surveys  were  completed,  indicating  a  prolate  spheroid,  the 
discussion  became  a  controversy.  Then  the  French  Academy 
resolved  to  send  expeditions  to  measure  two  meridian  arcs 
that  would  definitely  settle  the  matter,  one  near  the  equator 
and  another  as  far  north  as  possible. 

Accordingly  two  parties  set  out  in  1735,  one  for  Lapland, 
the  other  for  Peru.  The  Lapland  expedition  measured  its 
base  upon  the  frozen  surface  of  a  river,  executed  its  triangu- 
lation  and  latitude  observations,  and  returned  in  two  years 
with  the  results  /  =  92  778  toises,  B  =  I °. 6221.  The  Peru- 
vian expedition  measured  two  bases,  executed  its  triangula- 
tion  and  latitude  work,  and  returned  in  seven  years  with  the 
results  7=176875  toises,  0=3°.  1176.  From  these  the 
values  of  the  length  of  one  degree  were  found,  and  then  the 
following  results  could  be  written: 

Arc.  Mean  Latitude.  Toises  in  One  Degree. 
Lapland                    N.  66°  20'  57438 

France  N.  49    22  57  060 

Peru  S.      i    34  56  728 

These  figures  decided  the  question.  Since  that  time  every 
one  has  granted  that  the  earth  is  an  oblate  spheroid  rather 
than  a  sphere  or  an  prolate  spheroid. 

Prob.  51.  From  the  above  data  compute  the  radius  of  curvature  for 
the  Lapland  arc  and  for  the  Peruvian  arc. 

52.  MEASUREMENT  OF  MERIDIAN  ARCS, 

The  general  principles  regarding  the  measurement  of  a 
meridian  arc  have  been  given  in  Art.  50,  but  it  is  now  to  be 
noted  that  the  successful  execution  of  the  work  demands 
accurate  instruments,  good  observers,  and  long-continued 
labor.  The  latitude  observations  are  now  made  by  the  zenith 
telescope  method  of  Art.  49,  the  bases,  angles,  and  azimuths 


MEASUREMENT    OF   MERIDIAN   ARCS. 


145 


are  measured  with  corresponding  precision,  while  the  adjust- 
ment by  the  Method  of  JLeast  Squares  reduces  the  residual 
errors  to  a  minimum.  In.  the  last  century  these  precise 
methods  were  unknown,  yet  the  results  deduced  gave  valu- 
able information  and  progress  was  constantly  made  in 
methods  of  observation  and  computation.  It  will  be  of 
historic  interest,  perhaps,  to  give  a  brief  account  of  the  firs^ 
meridian  arc  measured  in  the  United  States. 

In  1763  the  proprietors  of  Pennsylvania  and  Maryland 
employed  two  astronomers  named  Mason  and  Dixon  to  locate 
the  boundary  lines  between  their  respective  possessions. 
This  occupied  several  years,  and  while  engaged  upon  it, 
Mason  and  Dixon  noted  that  several  of  the  lines,  particularly 
the  one  between  Maryland  and  Delaware,  were  well  adapted 
to  the  determination  of  the  length  of  a  degree,  being  on  low 
and  level  land,  and  deviating  but  little  from  the  meridian. 
Representing  this  to  the  Royal  Society  of  London,  of  which 
they  were  members,  they  received  tools  and 
money  to  carry  on  the  work.  The  measured 
lines  are  shown  in  the  annexed  sketch.  AB 
is  the  boundary  between  Delaware  and 
Maryland,  about  82  miles  long  and  making 
an  angle  of  about  four  degrees  with  the 
meridian;  BD  is  a  short  line  running  nearly 
east  and  west;  CD  andVW  are  meridians 
about  five  and  fifteen  miles  in  length  respec- 
tively;  CP  is  an  arc  of  the  parallel,  the  same 
in  fact  as  that  of  the  southern  boundary  of 
Pennsylvania.  In  1766  Mason  and  Dixon 
set  up  a  portable  astronomical  instrument  at 
A,  the  southwest  corner  of  the  present  State  of  Delaware, 
and  by  observing  equal  altitudes  of  certain  stars,  determined 
the  local  time  and  the  meridian,  after  which  the  azimuth  of 
the  line  AB  was  measured,  and  the  latitude  of  A  found  by 
observing  the  zenith  distances  of  several  stars  as  they  crossed 


146  SPHERICAL   GEODESY.  VI. 

the  meridian.  At  ^V,  a  point  in  the  forks  of  the  river 
Brandywine,  the  zenith  distances  of  the  same  stars  were  also 
measured,  from  which  it  was  easy  to  find  the  latitude  of  Nt 
and  the  difference  of  latitude  between  A  and  N.  In  1768  they 
made  the  linear  measurements  by  means  of  wooden  rectan- 
gular frames  20  feet  in  length.  All  the  lines  had  in  previous 
years  been  run  in  the  operation  for  establishing  the  boun- 
daries, and  along  each  of  them  "  a  vista"  cut,  which  "  was 
about  eight  or  nine  yards  wide,  and,  in  general,  seen  about 
two  miles,  beautifully  terminating  to  the  eye  in  a  point." 
Toward  this  point  they  sighted  the  wood  frames,  made  them 
truly  level  and  noted  the  thermometer  in  order  to  correct  for 
the  effect  of  temperature.  Through  the  swamps  they  waded 
with  the  wooden  frames,  but  across  the  rivers  they  found  the 
distance  by  a  measured  base  and  triangle. 

The  results  of  this  field  work,  as  sent  to  England  in  1768, 
were  as  follows:  latitude  of  A  =  38°  27'  34",  latitude  of 
N=  39°  56'  19",  azimuth  of  AB  at  A  =  176°  16'  30",  angle 
BDC—gf  2f  30",  ^  =  434011.6  feet,  BD  =  1489.9 
feet,  DC  =  26608.0  feet,  PN  =  78  290.7  feet,  DC  and  PN 
being  true  meridians  while  CP  was  an  arc  of  the  parallel. 

From  these  results  the  difference  of  latitude  between  A 
and  N  is  0  =  I°.479I7.  To  find  the  linear  distance  /,  an 
approximate  value  of  the  radius  of  the  earth  was  assumed 
and  each  of  the  measured  lines  projected  upon  the  meridian 
AN'  by  arcs  of  parallels  NN',  PP' ,  etc.  Thus  were  found 
^#=433078.8  feet,  £'£>'  =  Sg.S  feet,  D' P'  =  26  608.0 
feet,  and  P'N'  =  78  290.7  feet,  whose  sum  is  /  =  538  067.3 
feet.  The  length  of  one  degree  of  the  meridian  now  is 

1/6  =  363  764  feet  =  68.894  miles, 
from  which  the  radius  of  curvature  is 

Rt  =  57.2958/70  =  3947.4  miles. 
These    are    the    final    results    of     the    measurement    of    the 


53.  THE  EARTH   AS  A   SPHERE,  147 

meridian  arc  made  by  Mason  and  Dixon;  they  are  now 
k.iown  to  be  too,  small, v  the  present  accepted  values  for  the 
mean  latitude  of  the  arc^being  68.984  miles  and  3  952.4 
miles,  but  in  view  of  the  primitive  methods  employed  it  is 
.surprising  that  the  agreement  is  so  close. 

During  the  fifty  years  following  1750  a  number  of  meridian 
arcs  were  measured,  one  in  South  Africa,  one  in  Italy,  one  in 
Hungary,  one  in  Lapland,  while  in  France  and  England 
geodetic  surveys  furnished  the  data  for  computing  other  arcs. 
Most  important  of  all  was  the  triangulation  executed  in 
France  and  Spain  about  1800  for  determining  the  length  of 
the  meter,  which  embraced  an  arc  of  ten  degrees  in  length. 
All  these  arcs  confirmed  the  conclusion  that  the  earth  is  not 
a  sphere,  but  an  oblate  spheroid  flattened  at  the  poles. 

Prob.  52.  Compute  the  length  of  a  quadrant  of  the  meridian  in 
meters,  using  the  results  of  Mason  and  Dixon  and  supposing  the 
earth  to  be  a  sphere. 

53.  THE  EARTH  AS  A  SPHERE. 

Although  the  earth  is  not  a  sphere  it  is  sufficient  in  many 
investigations  to  regard  it  as  such,  since  the  amount  of 
flattening  at  the  poles  is  not  large.  In  fact,  if  the  earth  is 
represented  by  a  globe  sixteen  inches  in  equatorial  diameter 
the  polar  diameter  would  be  15.945  inches,  so  that  the  differ- 
ence between  the  two  diameters  would  not  be  perceptible  to 
the  eye.  The  question  now  arises  as  to  what  value  shall  be 
taken  for  the  radius  of  the  earth  and  what  is  the  mean  length 
of  a  degree  of  latitude  on  its  surface.  This  question  cannot 
be  answered  without  anticipating  to  a  certain  extent  some  of 
the  conclusions  of  the  next  chapter. 

The  mean  length  of  a  degree  of  latitude  is  the  mean  of  the 
lengths  of  all  the  degrees  from  the  equator  to  the  poles,  or 
one-ninetieth  of  the  elliptical  quadrant.  The  value  adopted 
for  the  quadrant  in  this  book  is 

q  —  10  ooi  997  meters  =  32  814886  feet, 


148  SPHERICAL   GEODESY.  VI. 

and  from  this  is  deduced  the  following  useful  table  of  mean 
lengths  of  arcs  on  the  earth's  meridian: 

One  degree  =   in  133         meters  =  364610  feet, 
One  minute  =        I  852.2     meters  =       6076.8  feet, 
One  second  =  30.87  meters  =        101.28  feet. 

The  mean  length  of  one  degree  may  also  be  stated  in  round 
numbers,  easy  to  remember,  as  69  statute  miles  or  60  nautical 
miles,  one  nautical  mile  thus  being  one  minute  of  latitude. 

The  mean  radius  of  the  earth,  considered  as  a  sphere,  must 
be  the  arithmetical  mean  of  all  the  radii  of  the  spheroid. 
This  is  evidently  the  same  as  the  radius  of  a  sphere  having  a 
volume  equal  to  the  volume  of  the  spheroid.  Let  a  be  the 
equatorial  and  b  the  polar  radius  of  the  oblate  spheroid, 
whose  accepted  values  are  6378278  and  6356654  meters 
respectively;  its  volume  is  ^ncfb.  Let  R  be  the  radius  of 
the  sphere  whose  volume  is  ^nR*.  Equating  these  values, 
there  is  found 

R  =  6371  062  meters  =  20  902  416  feet, 
or,  in  round  numbers, 

R  =  6371  kilometers  —  3  959  statute  miles, 
for  the  mean  radius  of  the  earth. 

This  mean  value  of  the  radius  is,  however,  incongruous 
with  the  above  mean  length  of  a  degree  of  latitude,  for  the 
quadrant  of  a  circle  corresponding  to  a  radius  of  6  371  kilo- 
meters is  nearly  six  kilometers  greater  than  the  true  elliptical 
quadrant.  In  certain  cases  it  might  be  more  logical  to  use 
the  radius  of  a  circle  whose  quadrant  is  equal  to  the  true 
quadrant;  this  requires  the  equation  \nR  =  10001997 
meters,  from  which 

R  —  6  369  kilometers  =  3  957  statute  miles, 
and   this  is   less  by  two   miles   than   the   mean   radius  of*  the- 
sphere.      This  discrepancy  is  unavoidable,  since  the  proper- 


54- 


LINES   ON  A   SPHERE. 


149 


ties  of  a  sphere  and  a  spheroid  are  not  the  same.  Thus  it  is 
impossible,  when  precisian  is  demanded,  to  regard  the  earth 
as  a  sphere. 

Prob.  53.  Taking  the  area  of  the  earth's  spheroidal  surface  as 
196  940  400  square  miles,  find  the  radius  of  a  sphere  having  the 
same  area. 


54.  LINES  ON  A  SPHERE. 

The  intersection  of  a  plane  and  a  sphere  is  always  a  circle. 
When  the  plane  passes  through  the  center  of  the  sphere  the 
circle  is  called  a  great  circle,  its  radius  P 

being  R  and  its  circumference  2nR. 
When  the  plane  does  not  pass  through  the  B> 
center  the  radius  of  the  circle  is  less  than 
R,  say  r,  and  its  circumference  is  2nr. 
All  great  circles  cut  out  by  planes  passing 
through  the  axis  of  the  earth  are  called 
meridians  and  these,  of  course,  converge 
and  meet  at  the  poles.  All  small  circles 
cut  out  by  planes  perpendicular  to  the 
axis  are  called  parallels.  Latitude  is 
measured  north  and  south  on  the  meridians 
from  the  equator  toward  the  poles,  while 
longitude  is  measured  east  and  west  on  the  parallels  from 
the  meridian  of  Greenwich. 

Using  the  mean  figures  of  the  last  Article,  one  minute  of 
latitude  corresponds  to  1852  meters  or  6077  feet.  One 
minute  of  longitude  on  the  equator  has  the  same  value,  but 
one  minute  of  longitude  on  any  parallel  circle  is  smaller  the 
nearer  the  circle  is  to  the  pole.  Thus  if  A  be  a  point  on  a 
parallel  whose  radius  AC  is  r,  and  whose  latitude  AOQ  is  0, 
and  if  R  be  the  radius  of  the  sphere,  then  r  =  R  cos0,  and 
accordingly  2nr  =  27r^-cos0,  that  is,  the  length  of  the 
parallel  circle  is  equal  to  the  length  of  a  great  circle  multiplied 


150  SPHERICAL   GEODESY.  VL 

by  the  cosine  of  the  latitude.  Hence  the  length  of  one 
degree  or  one  minute  of  longitude  at  any  latitude  is  found  by 
multiplying  the  values  of  the  last  Article  by  the  cosine  of  the 
latitude.  Thus,  using  I  852  meters  or  6  077  feet  for  the 
length  of  one  minute  at  the  equator,  the  length  of  one 
minute  of  longitude  at  latitude  40°  is  I  419  meters  or  4655 
feet,  while  at  latitude  80°  it  is  322  meters  or  I  055  feet. 

The  above  figure  shows  two  orthographic  projections  of  the 
meridians  and  parallels  of  a  sphere,  the  first  being  a  projec- 
tion on  a  plane  through  the  axis,  and  the  other  a  projection 
on  the  plane  of  the  equator.  The  parallels  appear  as  straight 
lines  in  the  first  diagram  and  as  circles  in  the  second. 

The  shortest  distance  on  the  surface  of  a  sphere  between 
any  two  points  is  along  an  arc  of  a  great  circle  joining  them. 
This  can  be  rigidly  demonstrated  by  establishing  a  general 
expression  for  the  length  of  a  line  on  the  spherical  surface 
and  making  it  a  minimum,  but  it  will  be  just  as  well  for  the 
student  to  satisfy  himself  of  the  truth  of  the  proposition  by 
actually  drawing  and  measuring  lines  on  such  a  surface.  As 
an  illustration,  the  distance  from  A  to  B  in  the  above  figure 
may  be  computed  by  the  route  ACB  along  the  parallel  and 
by  the  route  APB  on  the  great  circle.  The  length  of  the 
first  is  TtR  cos0  and  that  of  the  second  is  nR  —  2cj)R.  Thus, 
if  0  be  45  degrees  or  \it  radians,  the  first  route  has  the  length 
2.22R  while  the  second  has  the  length  1-57^.  In  like 
manner  the  distance  from  A  to  C  is  i.nR  along  the  parallel, 
but  I.O$R  along  a  great  circle  passing  through  A  and  C. 

The  azimuth  of  a  line  on  a  sphere  is  estimated,  as  in  a 
plane,  from  the  south  around  through  the  west;  thus  the 
northward  azimuth  of  all  meridians  is  180  degrees.  As  all 
meridians  converge  at  the  poles  the  back  azimuth  of  an 
oblique  line  is  not  equal  to  its  front  azimuth  plus  1 80°.  A 
great  circle  passing  through  C  with  an  east  and  west  direction 
at  that  point  cuts  the  neighboring  meridians  at  different 
angles  and  finally  crosses  the  equator  and  attains  the  same 


55.  ANGLES,   TRIANGLES,    AND    AREAS.  151 

southern  latitude  as  C  on  the  opposite  side  of  the  sphere. 
All  the  meridians  cut  the  equator  at  right  angles,  but  they  cut 
other  parallels  at  smaller  Angles.  An  oblique  line  crossing  all 
meridians  at  the  same  angle  is  of  a  spiral  nature  and  is  called 
a  loxidrome. 

Prob.  54.  What  part  of  the  surface  of  a  sphere  is  north  of  north 
latitude  60  degrees  ? 

55.  ANGLES,  TRIANGLES,  AND  AREAS. 

A  spherical  angle  is  the  plane  angle  between  the  tangents 
to  the  arcs  of  the  great  circles  at  their  point  of  intersection; 
thus  the  spherical  angle  BAG  is  the  same 
as  the  plane  angle  bAc.  When  a  horizontal 
angle  is  measured  at  a  station  A  on  the  sur- 
face of  the  earth,  the  limb  of  the  instrument 
is  made  level  or  tangent  to  the  spherical 
surface,  and  hence  when  pointing  is  made 
upon  B  and  C  the  plane  angle  bAc  is  the 
result  of  the  work.  If  the  triangle  be  of 
sufficient  size  it  will  be  found  that  the  sum 
of  the  three  measured  angles  is  greater  than  180  degrees. 

A  spherical  triangle  is  one  included  by  three  arcs  of  great 
circles.  It  is  a  well-known  geometrical  theorem  that  the 
sum  of  the  angles  of  a  spherical  triangle  is  greater  than  two 
right  angles,  and  that  the  excess  above  two  right  angles  bears 
the  same  ratio  to  a  right  angle  as  the  area  of  the  triangle 
bears  to  the  area  of  the  tri-rectangular  triangle.  The  tri- 
rectangular  triangle,  shown  by  PQO  in  the  figure  of  the  last 
Article,  is  one-eighth  of  the  surface  of  the  sphere  or  %7rR*. 
Thus  from  the  theorem  the  spherical  excess  is  given  by 

Excess  in  right  angles  =  area  of  triangle/£7r^J, 
or,  since  there  are  90  X  60  X  60  seconds  in  a  right  angle, 
Excess  in  seconds  —  648000  area/7rtfa.  (55) 


I$2  SPHERICAL   GEODESY.  VL 

Taking  for  R  the  mean  value  of  the  radius  of  the  earth  con- 
sidered as  a  sphere  (Art.  53),  this  becomes 

Excess  in  seconds  =  area  in  square  kilometers/197  *     ,, 
=  area  in  square  miles/76, 

which  are  convenient  approximate  rules  for  practical  use. 
Thus  a  triangle  has  one  second  of  spherical  excess  for  each 
197  square  kilometers  or  76  square  miles  of  area. 

The  same  rule  applies  to  quadrilaterals  or  polygons  on  the 
earth's  surface  bounded  by  great  circles,  the  word  excess 
meaning  the  excess  of  the  sum  of  the  interior  spherical  angles 
over  the  theoretic  sum  for  a  plane  figure.  Thus  a  polygon 
or  triangle  of  the  size  of  the  State  of  Connecticut  has  a 
spherical  excess  of  about  64  seconds;  this  amount  is  rarely 
exceeded  in  the  triangles  of  geodetic  triangulations  and  is 
usually  much  smaller. 

A  geodetic  triangle  is  necessarily  small  since  the  stations 
must  be  intervisible,  and  hence  its  curved  surface  does  not 
sensibly  differ  in  area  from  that  of  the  plane  triangle  formed 
by  lines  equal  in  length  to  the  spherical  arcs.  These  are  the 
distances  computed  from  the  triangulation  work,  and  the 
corresponding  plane  angles  are  found  by  subtracting  one-third 
of  the  spherical  excess  from  each  spherical  angle.  For 
instance,  let  two  sides  of  a  triangle  be  a  =  36  440  meters, 
b  =^  23  700  meters,  and  their  included  angle  (7  =  49°  05'; 
then  the  area  is  ^ab  s'mC  =  326.3  square  kilometers,  and  by 
(55)'  the  spherical  excess  is  oi".66. 

It  will  be  seen  later  that  the  above  equation  (55)  is  directly 
applicable  to  triangles  on  a  spheroid  by  taking  for  R  the 
radius  of  the  sphere  osculatory  to  the  spheroid  at  the  center 
of  gravity  of  the  triangle.  In  many  common  cases,  however, 
the  rough  rules  of  (55)'  will  give  the  spherical  excess  correctly 
to  hundredths  of  a  second. 

The  area  of  a  zone  of  the  sphere  bounded  by  the  parallel 
circles  whose  latitudes  are  Z,  and  Za  is  easily  derived.  The 


56. 


LATITUDES,    LONGITUDES,   AND   AZIMUTHS. 


153 


differential  expression  is  2nrRdL,  where  r  is  the  radius  of  the 
parallel  and  R  that  of  thevsphere.  But  r  =  R  cosZ,  and  hence 

A  =  27tR*~l  cosLcfL  =  2  7i R*  (sin Z,  —  sinZj) 

is  the  area  of  the  zone  between  the  upper  latitude  Z,,  and  the 
lower  latitude  L^  i  Thus  to  find  the  area  between  latitude 
30°  and  the  equator,  Za  =  30°  and  Ll  =  o°,  whence  A  =  nR* 
or  one-fourth  of  the  surface  of  the  sphere. 

The  area  of  a  trapezoidal  degree,  that  is,  of  a  surface 
bounded  by  two  parallels  one  degree  apart  and  by  two 
meridians  one  degree  apart,  may  be  readily  deduced  from  the 
last  equation  and  will  be  found  o. 000304167  ^2  cosZ,  in  which 
L  is  the  middle  latitude  of  the  trapezoid.  Thus,  taking  R  = 
3  959  miles  and  L  =  45°,  the  area  of  the  trapezoidal  degree 
is  3  376  square  miles. 

Prob.  55.  In  a  spherical  triangle  two  angles  are  observed  to  be 
79°  03'  4i".93  and  59°  35'  44".38,  and  the  included  side  is  23  700 
meters.  Compute  the  spherical  excess  and  find  the  other  spherical 
angle. 

56.  LATITUDES,  LONGITUDES,  AND  AZIMUTHS. 

Let  A  and  B  be  two  points  on 
the  surface  of  the  sphere,  L  and  M 
being  the  latitude  and  longitude  of 
A,  and  L'  and  M'  those  of  B.  The 
latitudes  are  estimated  northward 
from  the  equator  and  the  longitudes 
westward  from  the  meridian  of 
Greenwich,  both  in  degrees,  min- 
utes, and  seconds  of  arc.  Let  a 
great  circle  connect  the  points  A 
and  B,  and  let  its  angular  length 
be  5.  Let  the  meridians  through 
A  and  B  be  produced  to  meet  at 
the  north  pole  P  and  to  cross  the  equator  at  Q  and  Q.' 


154  SPHERICAL   GEODESY.  VI. 

The  azimuth  of  AB  is  the  angle  QAB,  which  is  called  Z, 
and  the  azimuth  of  BA  is  the  obtuse  angle  Q '  BA,  which 
is  called  Z' .  Let  the  latitude  and  longitude  of  A  be  given, 
together  with  the  length  and  azimuth  of  AB.  It  is  re- 
quired to  find  the  latitude  and  longitude  of  B  and  the  azi- 
muth of  BA. 

In  the  spherical  triangle  ABP  the  side  PA  is  90°  —  Z,  the 
side  PB  is  90°  —  L ',  and  the  side  AB  is  S;  the  angle  A  is 
180°  — Z,  the  angle  B  is  Z'  —  180°,  and  the  angle  P  is 
M'  —  M.  Writing  the  formula  of  spherical  trigonometry  for 
the  cosine  of  PB  in  the  notation  here  used,  it  becomes 

cos  (90°-Z')=cos  (9o°-Z)  cosS+sin  (90°-Z)  sin  S  cos  (i8o°-Z), 
which  reduces  immediately  to 

sinZ'  =  sinL  cosS  —  cosL  s'mS  cosZ,  (56) 

and  the  latitude  L'  is  hence  expressed  in  terms  of  known 
quantities. 

In  the  same  triangle,  using  the  theorem  that  the  sines  of 
the  sides  PB  and  AB  are  proportional  to  the  sines  of  the 
opposite  angles, 

sin  (90°  -  L')/s'mS  =  sin  (i  80°  -  Z)/sin  (Mf  -  M), 
or  sin  (Mf  —  M)  =  sinS  sinZ/cosZ',  (56)' 

from  which  the  longitude  Mf  can  be  computed  after  the  lati- 
tude L'  has  been  found. 

To  deduce  Z'  one  of  the  formulas  known  as  Napier's 
analogies  will  be  most  convenient  in  numerical  work,  namely, 

tan  \(A  +  £)/cotK  =  cos|(/>£-  PA)/cos%(PB  +  PA), 
and,  reducing  this  to  the  notation  in  hand,  it  becomes 
cot  £(Z'-  Z)  =  tan  \(M'  -  M}  sin  \(L  -f  Z')/cos  \(L'  -  L),  (56)" 
from  which  the  back  azimuth  Zf  can  be  computed. 

These  formulas  apply  to  a  spherical  arc  of  any  size  on  any 
sphere.  For  example,  let  L  =  40°  45^  and  M  =  73°  58', 
these  being  values  for  New  York  City,  and  let  it  be  required 


56.  LATITUDES,    LONGITUDES,   AND   AZIMUTHS.  155 

to  find  L',  Mf,  and  Z' ,  for  a  point  whose  angular  distance  is 
5=35°  and  whose  azimuth  at  New  York  is  Z  =  90°.  From 
formulas  (56)  and  (£6)' 

log  s'mL'  =  1.72812,          Lf  =     32°  19', 
log  sin  (M'-M)=  1.83172,        M'  =  u6'°43/, 

which  indicates  that  the  point  is  located  in  the  vicinity  of 
San  Diego,  California.     To  find  Z'  the  formula  (56)"  gives 
log  coti(Z  -  Z')  ^  1.36852,        Z  =  243°  42', 

which  shows  that  if  a  great  circle  be  drawn  between  the  two 
places  the  direction  of  this  is  due  west  at  New  York,  but  at 
San  Diego  its  direction  is  N.  64°  E.  As  the  earth  is  not  a 
sphere,  these  results  may  be  a  degree  or  more  in  error. 

Then  a  line  AB  runs  due  north  its  azimuth  Z  is  180°  ;  th  :i 
( .-6)  reduces  to  Z/  =  Z,  +  S,  and  (56)'  gives  Mf  =  M,  vh. 
(  ')"  shows  thit  Z'  =  Z  +  180°  =  o°.  If  it  runs  due  L.  irjj 
r  :.,  c°  ;  then  (56)  gives  L'  =  L  -  S  and  (56)'  gives  M'  •-  M, 
>  !i  "c  (56)"  shows  that  Z'  =  Z  +  180°  =  180°.  If  the  two 
;,  L.tJ  arc  on  the  same  parallel  of  latitude,  then  L'  =  L  and 
S  h  th.ir  angular  distance  on  a  great  circle. 

For  most  geodetic  triangles  the  lengths  of  the  sides  are  so 
small  compared  with  the  radius  of  curvature  R  that  it  is  suffi- 
cient to  take  cosS  =  I  and  sinS  =  l/R,  where  /  is  the  length 
of  the  arc  or  chord  joining  A  and  B.  Then  the  above 
formulas  may  be  directly  applied  to  such  triangles,  and  in 
Art.  64  it  will  be  shown  how  they  are  further  simplified. 

Prob.  56.  Given  L  =  40°  36'  22^.452,  M  =  75°  22'  5i".i5o, 
Z=  193°  56'  28". i,  and  I—  1726.60  meters.  Taking  ^  =  6371 
kilometers,  compute  latitude  L ',  longitude  J/',  and  azimuth  Z' '. 


156  SPHEROIDAL   GEODESY.  VII, 


CHAPTER  VII. 
SPHEROIDAL  GEODESY. 

57.  PROPERTIES  OF  THE  ELLIPSE. 

Since  an  oblate  spheroid  is  generated  by  the  revolution  of 
an  ellipse  about  its  minor  axis,  the  equator  and  all  the  sec- 
tions of  the  spheroid  parallel  to  the  equator  are  circles,  and 
all  sections  made  by  planes  passing  through  the  axis  of 
revolution  are  equal  ellipses.  Let  a  and  b  represent  the 
lengths  of  the  semi-major  and  semi-minor  axes  of  this 
meridian  ellipse,  which  are  the  same  as  the  semi-equatorial 
and  semi-polar  diameters  of  the  spheroid;  when  the  values  of 
a  and  b  have  been  found  all  the  other  dimensions  of  the 
ellipse  and  the  spheroid  become  known.  It  is  necessary  first 
to  deduce  several  equations  expressing  the  properties  of  the 
ellipse,  and  then  by  discussing  them  in  connection  with  the 
results  of  measurements  of  meridian  arcs  the  form  and  size  of 
the  spheroid  is  to  be  found. 

The  eccentricity  of  an  ellipse  is  the  ratio  of  the  distance 
between  focus  and  center  to  the  semi-major  axis,  and  the 
ellipticity  is  the  ratio  of  the  flattening  of  one  pole  to  the 
semi-major  axis.  Let  e  be  the  eccentricity  and  f  be  the 
ellipticity,  then 


-  b*  a-  b 


The  relation  between  these  two  fractions  is 


57- 


PROPERTIES   OF  THE   ELLIPSE. 


'57 


and  b  may  be  expressed  in  terms  of  e  and/ in  two  ways, 


b  = 


—  e 


=  *(!-/). 


Thus  two  quantities  determine  an  ellipse;  those  generally 
used  are  a  and  ^,  and  When  these  have  been  found  b  and  f 
are  known. 

The  equation  of  the  ellipse  referred  to  the  rectangular  axes 
QQ  and  PP  is  ay  +  £V  =  #2£2,  in  which  j/  and  x  are  the 
ordinate  and  abscissa  of  any  point  A.  Let  L  be  the  latitude 
of  A,  that  is,  the  angle  ABQ,  and  let  it  be  required  to  find 


the  relation  between  x  and  L.     At  A  draw  the  tangent  AT, 
and,  since  the  angle  A  TB  is  90°  —  L, 


tan(9o 


whence 


tfx 
y  =  —^  tanZ. 


Inserting  this  in  the  rectangular  equation  of  the  ellipse,  and 
replacing  &  by  its  value  a\i  —  e9),  there  is  found,  after 
reduction, 

x  —  a  cosL/(i  —  e*  sin'Z)*,  (57) 

which  is  an  equation  of  the  ellipse  in  terms  of  the  variables 
x  and  L.  If  L  =  o°  then  x  =  a,  and  if  L  —  90°  then  x  =  o. 

To  find  the  radius  of  curvature  of  the  ellipse  at  A  the  rect- 
angular equation  of  the  ellipse  may  be  written  in  the  form 

y  =  *-(*>  -  *•)'. 


158  SPHEROIDAL   GEODESY.  VII. 

and  the  first  and  second  derivatives  of  y  with  respect  to  x  are 
dy  _  bx  d*y  ab 

d*  ~~       '  a(a    -  *')*'          ~dx*  =      "  (a*  -  x*f 

and  then,  replacing  tf  by  a\\  —  e*)  and  x  by  its  value  from 

(57), 


which   is  the   required   radius  of  curvature.      If  L  =  o°   then 
RI  =  &/a\  if  L  =  90°  then  ^  =  c?/b. 

To  find  the  length  of  an  arc  of  the  ellipse  the  differential 
element  R^dL  is  to  be  integrated  between  the  limits  o  and  L. 
This  furnishes  an  elliptic  integral  which  cannot  be  evaluated 
except  by  a  series;  thus 

/  _  a(i  -  /)  CL(i  +  $S  sin'Z  +  Y^4  sin4/-  +•  •  -X£» 

I/O 

the  integration  of  which  gives 

/  =  a(i  -  e*)l(i  +  t*a  +  ll^4  +  ...)/—(f'a  +  IS-'*  +  -)  sin*£  +  (^V4  +  ...)  sin4/.-...]     (57)" 

for  the  length  of  a  meridian  arc  from  o°  to  L°.      If  L  =  90° 
=  %TT,  then 

/,  =  i«(i  -i^--TV'-rf^'--  •••). 

which  is  the  length  of  a  quadrant  of  the  ellipse. 

Prob.  57.  Given  e  =  0.082271,  and  a  =  6  378  206  meters,  to  com- 
pute the  length  of  the  quadrant  .to  the  nearest  meter. 

58.  DISCUSSION  OF  MERIDIAN  ARCS. 

Since  a  spheroid  is  determined  by  the  two  elements  a  and 
e  of  the  generating  ellipse  two  equations  are  required  to  find 
their  values.  These  may  be  established  by  the  discussion  of 
two  meridian  arcs  in  different  latitudes.  Let  /,  and  /„  be 
their  lengths,  0l  and  #a  their  amplitudes  or  the  number  of 
degrees  of  latitude  between  their  northern  and  southern  ends, 
Z,,  and  Lt  the  latitudes  of  their  middle  points,  an  1  Rl  and  ^?t 


58. 


DISCUSSION   OF   MERIDIAN   ARCS. 


159 


the  radii  of  curvature  at  these  points.      Regarding  these  arcs 
as  arcs  of  circles,  their  radii  of  curvature  are 

1804 


but  considering  the  middle  points  as  lying  upon  the  circum- 
ference of  an  ellipse  their  radii,  as  given  by  (57)',  are 

R   -      -fCiJiLfV  R  -          a(l  ~  ^ 

1         ~     (  ,  J         •       2   T     \§'  2  /  9         •       1   T    \$* 

Equating  the  values  of  R^  and  also  the  values  of  R^  there  are 
found  two  equations  whose  solution  gives 

I  —  u 


i8o/,(i  -  S  sin'Z,)'       i8o/,(i  -  /  sin'Z,)* 

" 


(59) 


by  which  the  eccentricity  of  the  ellipse  and  its  major  axis 
may  be  computed  from  the  data  of  two  measured  meridian 
arcs. 

It  is  plain  that  these  elements  wiii  be  most  accurately 
determined  when  one  arc  is  as 
near  the  pole  as  possible  while  the 
other  is  at  the  equator.  These 
conditions  exist  in  the  Lapland 
and  Peruvian  arcs  (Art.  51),  the 
results  of  which  became  known 
about  1745.  The  data  for  these  arcs  are  as  follows: 


Peruvian  Arc. 
/,  =  176  875.5  toises 


Lapland  Arc. 

A  —        92  777-98  toises 

0,  =          i°  37'  V-57  0*  =        3°  °7'  °3"-46 

Z,  =  +  66°  20'  io".o5,  Z,  =  —  i°  31'  oo".34 

Reducing  the  amplitudes  to  degrees,  and  substituting  in  (59), 
there  results 


*  =  0.00643506,      whence 


—  0.080219, 

CIYIL  ENGINEERING 

U.  ofC. 
ASSOCIATION  LIBRARY 


l6o  SPHEROIDAL   GEODESY.  VII. 

and  then  substituting  the  value  of  e*  in  either  of  the  values  of 
a  there  is  found  a  —  3  271  652  toises.  These  two  values 
completely  determine  the  ellipse  and  the  oblate  spheroid 
generated  by  it.  Then,  from  the  expressions  of  the  last 
Article,  /=  0.003223,  and  the  length  of  the  quadrant  is 
5  130817  toises,  or  10000  150  meters. 

It  is  often  customary  to  state  the  values  of  e  and  f  as  vulgar 
fractions,  since  thus  a  clearer  idea  of  the  oblateness  of  the 
spheroid  is  presented.  For  this  case  the  rough  values  are 

1  f  l 

~    12.$'  ~    310.3' 

or  the  distance  of  the  focus  of  the  ellipse  from  the  center  is 
T^.-g-th  and  the  flattening  at  one  of  the  poles  is  .^.^th  of  the 
equatorial  radius. 

About  the  year  1745  the  results  of  the  surveys  instituted 
by  the  French  Academy  became  known ;  these  have  been 
given  in  Art.  51  in  toises.  The  length  of  one  degree  of  lati- 
tude is  //#,  if  0  be  in  degrees,  and  thus  these  data  give  every- 
thing necessary  for  computing  u,  e,  and  a  from  the  above 
formulas.  From  these  three  arcs  three  computations  were 
made  by  the  above  method,  and  these  gave  results  about  as 
folljws  for  the  ellipticity  of  the  spheroid: 

From  Lapland  and  French  arcs,  f  — 
From  Lapland  and  Peruvian  arcs,  f  = 
From  French  and  Peruvian  arcs,  f  =  ¥^T. 

Now,  if  the  earth  be  a  spheroid  of  revolution,  and  if  the 
measurements  be  precise,  these  values  of  the  ellipticity  should 
be  the  same.  Since,  however,  they  disagree  the  conclusion 
was  easy  to  make  either  that  the  assumption  of  the  spheroid 
was  incorrect  or  that  the  surveys  were  lacking  in  precision. 

After  the  year  1750,  when  the  results  of  the  Lapland  and 
Peruvian  arcs  had  become  known,  great  interest  was  mani- 
fested in  securing  additional  data  by  the  measurement  of 


58. 


DISCUSSION    OF   MERIDIAN   ARCS. 


161 


other  meridian  arcs  in  order  to  determine  whether  or  not  the 
earth  was  a  true  ellipsoid  of  revolution.      The  following  table 

MERIDIAN    ARCS. 


No. 

Locality  of  Arc. 

Middle  Latitude. 

Length  of 
One  Decree. 

I 

Lapland, 

+   66°  20' 

Toises. 
57405 

2 

Holland, 

+    52    °4 

57H5 

3 

France, 

+  49   23 

57  O74i 

4 

Austria, 

+  48   43 

57086 

5 

France, 

+  45    43 

57034 

6 

Italy, 

+  43   OI 

56979 

7 

Pennsylvania, 

+  39    12 

56888 

8 

Peru, 

-     i    34 

56753 

9 

Cape  of  Good  Hope, 

-  33    18 

57037 

gives  the  results  of  nine  arcs  which  were  measured  during  the 
eighteenth  century  and  discussed  by  Laplace  in  1799.  For 
this  purpose  he  took  the  expression  for  the  radius  of  curva- 
ture given  in  (57)',  developed  it  by  the  binomial  formula,  and 
divided  it  by  iSo/TT,  thus  obtaining 


d  = 


- 

I  oO 


sin'Z 


as  an  expression  for  the  length  of  one  degree  of  latitude.  It 
thus  appeared  that  the  length  of  a  degree  could  be  expressed 
by 

d  =  M  +  N  sin'Z  +  P  sin'Z  +  .  .  ., 

in  which  M  =  na(i  -  *')/i8o,   N  =  %e'M,    P  =  -1/  e4M, 
and  Laplace  in  discussing  the  above  data  concluded  that  it 
was    unnecessary  to   retain   the  term   containing  P  since  its 
value  is  small.      Accordingly  he  wrote 

d  =  M  +  N  sin'Z, 
and  then  proceeded  to  find  probable  values  of  M  and  N  from 


162  SPHEROIDAL   GEODESY.  VII. 

the  nine  observations  of  the  above  table,  and  from  these  to 
deduce  the  values  of  a  and  e. 

At  that  time  the  Method  of  Least  Squares  was  unknown, 
but  Laplace  wrote  the  nine  observation  equations,  and  then 
used  the  two  conditions  that  the  algebraic  sum  of  the  errors 
should  be  zero  and  that  the  sum  of  the  same  errors  all  taken 
positively  should  be  a  minimum.  He  thus  obtained  two 
resulting  equations  from  which  he  found  M  =  56  753  toises, 
N  =  613.1  toises,  and  accordingly 

d  =  56753  +  613.1  sin'Z 

is  an  empirical  formula  for  the  length  of  one  degree.  From 
these  values  of  J/and  -/V  he  found  e*  =  2N/$M  =  0.007202, 
and  then  f  =  ^g. 

The  last  step  in  Laplace's  investigation  is  the  comparison 
of  the  observed  values  of  the  lengths  of  the  degrees  with 
those  computed  from  his  empirical  formula.  For  the  Lapland 
arc,  for  instance,  observation  gives  d  =  57  405  toises,  while 
the  formula  gives  d  =  57  267  toises,  the  difference,  or  residual 
error,  being  138  toises,  a  distance  equal  to  nearly  900  feet, 
or  to  nearly  9  seconds  of  latitude.  These  errors,  says 
Laplace,  are  so  great  that  they  cannot  result  from  the  in- 
accuracies of  surveys,  and  hence  it  must  be  concluded  that 
the  earth  deviates  materially  from  the  elliptical  figure. 

At  the  beginning  of  the  nineteenth  century  it  was  the  pre- 
vailing opinion  among  scientists,  founded  on  investigations 
similar  to  that  of  Laplace,  that  the  contradictions  in  the  data 
derived  from  meridian  arcs,  when  combined  on  the  hypothesis 
of  an  oblate  spheroidal  surface,  could  not  be  attributed  to 
the  inaccuracies  of  surveys,  but  must  be  due  in  part,  at  last, 
to  deviations  of  the  earth's  figure  from  the  assumed  form. 
This  conclusion,  although  founded  on  data  furnished  by  sur- 
veys that  would  nowadays  be  considered  rude,  has  been 
confirmed  by  all  later  investigations,  so  that  it  can  be  laid 
down  as  a  demonstrated  fact  that  this  earth  is  not  an  oblate 


58.  DISCUSSION   OF   MERIDIAN   ARCS.  163 

spheroid.  Yet  it  must  never  be  forgotten  that  the  actual 
deviations  from  that  form  are  very  small  when  compared  with 
the  great  size  of  the  glob*  itself.  In  some  of  the  practical 
problems  into  which  the  shape  of  the  earth  enters  it  is  suffi- 
cient to  regard  it  as  a  sphere,  in  many  others  a  spheroid  must 
be  used,  while  in  only  a  few  cases  is  it  required  to  regard  the 
deviation  from  the  spheroidal  form.  Now  it  was  agreed  by 
all  in  the  early  part  of  the  nineteenth  century,  that  for  the 
practical  purposes  of  mathematical  geography  and  geodesy  it 
was  highly  desirable  to  determine  the  elements  of  an  ellipse 
agreeing  as  closely  as  possible  with  the  actual  meridian  section 
of  the  earth,  or,  in  other  words,  that  the  most  probable 
spheroid  should  be  deduced  from  the  data  of  observation. 
This  search  after  the  most  probable  ellipse  resulted  in  the 
discovery  by  Legendre,  in  1805,  of  the  Method  of  Least 
Squares,  and  the  first  problem  to  which  this  method  was 
applied  was  a  discussion  of  the  elements  of  the  ellipse  result- 
ing from  five  portions  of  the  French  meridian  arc. 

Important  geodetic  work  was  carried  on  in  France  and 
Spain  by  Delambre  and  Mechain  for  determining  the  length 
of  the  meter,  which,  with  the  accompanying  office  work, 
lasted  from  1792  to  1807.  The  meridian  arc  embraced  an 
amplitude  of  nearly  ten  degrees,  and  the  methods  for  the 
measurement  of  bases  and  angles  were  greatly  improved,  in 
fact  approaching  for  the  first  time  to  modern  precision.  The 
results  were  combined  with  those  of  the  Peruvian  arc  to  find 
the  eccentricity,  and  this  gave  for  the  ellipticity  -g^  and  for 
the  quadrant  5  130740  toises.  This  was  equivalent  to 
looooooo  meters,  since  by  the  French  law  the  meter  had 
been  defined  to  be  one  ten-millionth  part  of  the  quadrant. 
It  is  now  known  that  this  length  of  the  quadrant  is  too  small 
by  nearly  2  ooo  meters  (Art.  60). 

Prob.  58.  Explain  how  the  method  of  least  squares  is  to  be  applied 
to  the  deduction  of  an  empirical  formula  for  the  length  of  one  degree 
from  the  data  in  the  above  table. 


164 


SPHEROIDAL  GEODESY. 


VII. 


59.  PLUMB-LINE  DEFLECTIONS. 

During  the  nineteenth  century  many  investigations  of  the 
size  and  shape  of  the  elliptical  meridian  of  the  oblate  spheroid 
were  made.  The  most  important  of  these  gave  the  results 
for  the  ellipticity  and  for  the  length  of  the  quadrant  which 
are  stated  in  the  following  table: 

DIMENSIONS   OF   SPHEROIDS. 


Year. 

By  whom. 

Ellipticity. 

Quadrant  in 
Meters. 

1810 

Delambre 

1/334 

10  000  000 

1819 

Walbeck 

1/302.8 

10  000  268 

1830 

Schmidt 

1/297.5 

10  ooo  075 

1830 

Airy 

1/299.3 

10  ooo  976 

1841 

Bessel 

1/299.2 

10  ooo  856 

1856 

Clarke 

1/298.1 

10  ooi  515 

I863 

Pratt 

1/295.3 

10  ooi  924 

1866 

Clarke 

1/295.0 

10  ooi  887 

1868 

Fischer 

1/288.5 

10  ooi  714 

1878 

Jordan 

1/286.5 

10  ooo  681 

1880 

Clarke 

1/293.5 

10  ooi  869 

Mcst  of  these  results  were  determined  by  a  discussion  of  the 
da*  a  of  several  meridian  arcs  by  the  Method  of  Least  Squares, 
ar//l  a  brief  explanation  is  now  to  be  given  as  to  how  such 
^inputations  are  made. 

The  principle  of  the  Method  of  Least  Squares  (Art.  3) 
equires  that  the  sum  of  the  squares  of  the  errors  of  observa- 
tion shall  be  rendered  a  minimum  in  order  to  give  the  most 
probable  values  of  the  observed  quantities.  The  first  inquiry 
then  is  as  to  where  the  errors  of  observation  in  a  meridian 
arc  lie;  are  they  in  the  linear  distance  /  or  in  the  angular 
amplitude  6  ?  The  error  in  a  linear  distance  that  is  computed 
from  a  good  triangulation  is  known  to  be  very  small,  say 


PLUMB-LINE   DEFLECTIONS.  165 


less  than  -^^nrV^-g-th  part  of  its  length  (Art.  26).  The  error 
in  an  observed  latitude  found  by  the  zenith-telescope  method 
cannot  exceed  half  a  second  (Art.  49).  Neither  of  these 
errors  can  account  for  even  a  small  part  of  the  discrepancy 
that  is  found  between  the  observed  and  computed  length  of 
a  degree  of  latitude  (Art.  59). 

Early  in  the  nineteenth  century  it  was  suspected  that  the 
cause  of  these  discrepancies  was  due  to  deflections  of  the 
plumb  lines  from  the  normal 
to  the  spheroid.  To  illus- 
trate let  the  sketch  represent 
a  very  small  part  of  a  meridian 
section  of  the  earth.  O  is  the 
ocean,  M  a  mountain,  and  A 
a  latitude  station  between 
them  ;  ece  is  a  part  of  the 
meridian  ellipse  coinciding  with  the  ocean  surface;  Ac  repre- 
sents the  normal  to  the  ellipse,  and  Ah,  perpendicular  to  Ac, 
the  true  level  for  the  station  A.  Now  owing  to  the  attraction 
of  the  mountain  M,  the  plumb  line  is  drawn  southward  from 
the  normal  to  the  position  AC,  and  the  apparent  level  is 
depressed  to  AH.  If  AP  be  parallel  to  the  earth's  axis,  and 
hence  pointing  toward  the  pole,  the  angle  PAh  is  the  latitude 
of  A  for  the  spheroid  eee;  but  as  the  instrument  at  A  can 
only  be  set  for  the  level  AH,  the  observed  latitude  is  PAH, 
which  is  greater  than  the  former  by  the  angle  hAH.  These 
differences  or  errors  are  usually  not  large,  rarely  exceeding 
ten  seconds,  yet  since  a  single  second  of  latitude  corresponds 
to  about  31  meters  or  101  feet,  it  is  evident  that  the  error 
due  to  these  plumb-line  deflections  may  be  very  great  com- 
pared with  any  accidental  error  in  the  measured  length  of 
the  meridian  arc. 

It  is  not  necessary,  of  course,  that  there  should  be  any 
plumb  bob  on  an  instrument  for  determining  latitude,  but 
whatever  affects  a  plumb  line  affects  the  level  bubble  by 


166  SPHEROIDAL  GEODESY.  VI  I. 

which  the  graduated  limb  is  made  horizontal.  Even  when  a 
sextant  is  used  the  effect  of  gravity  upon  the  mercury  in  the 
artificial  horizon  may  make  its  surface  deviate  from  parallelism 
to  the  tangent  plane  of  the  spheroid.  Thus,  the  term 
plumb-line  deflection  means  really  the  elevation  or  depres- 
sion of  the  astronomical  horizon  with  reference  to  the  plane 
tangent  to  the  spheroid.  Astronomical  latitude  is  determined 
with  reference  to  a  vertical  line  at  the  place  of  observation 
(Art.  40),  but  geodetic  latitude  is  with  reference  to  a  normal 
to  a  spheroid  at  that  point,  and  the  difference  of  these  is 
called  the  plumb-line  deflection.  A  plumb-line  direction  and 
an  astronomical  latitude  are  real  things,  but  a  normal  to  a 
spheroid  and  the  corresponding  geodetic  latitude  are  artificial 
things,  and  hence  a  plumb-line  deviation  depends  upon  the 
particular  spheroid  to  which  it  is  referred. 

In  deducing  the  elements  of  an  ellipse  from  the  data  of 
meridian  arcs,  the  lengths  are  hence  to  be  taken  as  without 
error,  and  the  sum  of  the  squares  of  the  errors  in  the  latitudes 
is  to  be  made  a  minimum.  For  this  purpose  let  /  be  the 
length  of  an  arc  and  6  its  amplitude  in  seconds;  the  radius  of 
curvature,  regarding  it  as  an  arc  of  a  circle,  is  206  26$l/8,  and 
equating  this  to  the  expression  for  the  radius  of  curvature 
given  by  ($?)'  there  is  found 

0  =  2062657(1  —  e*  sin'Z)f/rf(i  -  e*), 

in  which  L  is  the  middle  latitude.  Now  if  L'  and  L"  be  the 
latitudes  of  the  north  and  south  ends  of  the  arc,  this  expres- 
sion becomes,  after  developing  the  parenthesis  and  neglecting 
powers  of  e  higher  than  the  second, 


+  206  265/(l  _   s  sin.z 

This  equation  contains  the  two  elements  a  and  e1  whose  values 
are  to  be  found,  while  the  other  quantities  are  the  data  of  the 
meridian  arc.  Now  let  v^  and  vt  be  corrections  to  be  applied 
to  the  observed  latitudes  L1  and  L"  ',  these  being  the  plumb- 


59.  PLUMB-LINE   DEFLECTIONS.  l6/ 

line  deviations  from  the  normals  to  the  spheroid ;  also  let  x 
be  a  correction  to  be  appHed  to  an  assumed  value  of  ay  and  y 
a  correction  to  be  applied  teo  an  assumed  value  of  e1.  Then 
this  equation  may  be  put  into  the  form 

vl  —  :>,  =  mx  -\-ny-\-p, 

where  m,  n,  and  /  are  known  functions  of  the  observed 
quantities.  Now  if  there  be  three  meridian  arcs,  each  having 
two  latitude  stations,  there  will  be  six  plumb-line  deviations; 
thus  for  the  first  arc  the  two  corrections  may  be  written 

vt  =  vz  -f-  mx  -f-  ny  -\-  p, 

v*  =  vt  , 

and  similarly  for  the  other  arcs.  If  the  left  members  be 
made  zero  these  are  the  six  observation  equations  which 
contain  the  five  unknown  quantities  z/a ,  z/4 ,  vt ,  x,  and  y. 
The  normal  equations  are  now  formed,  and  their  solution 
gives  the  most  probable  values  of  x  and  j,  from  which  those 
of  a  and  e*  are  found,  and  also  the  most  probable  values  of 
the  plumb-line  deviations. 

Such  is  a  brief  outline  of  the  process  of  determining  the 
size  of  the  earth  from  several  measured  meridian  arcs.  In 
practice  the  numerical  work  is  abbreviated  by  using  the 
method  of  correlates  (Art.  78),  but  even  then  is  very  lengthy, 
several  weeks  being  required  to  form  and  solve  the  normal 
equations  when  many  arcs  are  used.  Each  arc,  moreover, 
generally  has  several  latitude  stations,  so  that  the  number  of 
observation  equations  is  more  than  twice  as  many  as  there 
are  meridian  arcs.  The  spheroid  thu^  deduced  is  the  most 
probable  one  that  can  be  derived  from  the  given  data,  for  the 
sum  of  the  squares  of  the  errors  in  the  latitudes  has  been 
made  a  minimum. 

Prob.  59.  Consult  Clarke's  investigation  of  1866  in  Comparison  of 
Standards,  published  by  the  British  Ordnance  Survey,  and  ascertain 
the  number  of  meridian  arcs  used,  the  number  of  normal  equations, 
and  the  greatest  values  of  the  plumb-line  deflections. 


168  SPHEROIDAL  GEODESY.  VIL 

60.  DIMENSIONS  OF  THE  SPHEROID. 

The  most  important  investigations  for  the  determination 
of  the  size  of  the  spheroid  are  those  made  by  Bessel  in  1841 
and  by  Clarke  in  1866.  The  data  employed  by  Bessel 
included  ten  meridian  arcs,  namely,  one  in  each  of  the 
countries  Lapland,  Denmark,  England,  France,  and  Peru, 
two  in  Germany,  and  two  in  India.  The  sum  of  the  ampli- 
tudes of  these  arcs  is  about  50.5  degrees,  and  they  include 
38  latitude  stations.  In  the  manner  briefly  described  above, 
there  were  written  38  observation  equations,  from  which  12 
normal  equations  containing  12  unknown  quantities  were 
deduced.  The  solution  of  these  gave  the  elements  of  the 
meridian  ellipse,  and  also  the  residual  errors  in  the  latitudes 
due  to  the  deflections  of  the  plumb  lines.  The  greatest  of 
these  errors  was  6". 45,  and  the  mean  value  2 ".64.  The 
spheroid  resulting  from  this  investigation  is  often  called  the 
Bessel  spheroid,  and  the  elements  of  the  generating  ellipse, 
Bessel's  elements;  the  values  of  these  will  be  given  below. 

In  1866  Clarke,  of  the  British  Ordnance  Survey,  published 
a  valuable  discussion,  which  included  a  minute  comparison 
of  all  the  standards  of  measure  that  had  been  used  in  the 
various  countries.  The  data  were  derived  from  six  arcs, 
situated  in  Russia,  Great  Britain,  France,  India,  Peru,  and 
South  Africa,  including  40  latitude  stations,  and  in  total 
embracing  an  amplitude  of  over  76°.  The  mean  value  of  the 
plumb-line  deflections,  or  latitude  errors,  was  found  to  be 
I ".42.  This  investigation  is  generally  regarded  as  the  most 
important  one  of  the  last  quarter  of  a  century,  and  the  values 
derived  by  it  as  more  precise  than  those  of  Bessel.  The 
Clarke  spheroid,  as  it  is  generally  called,  has  been  used  in 
most  of  the  geodetic  work  done  in  America  since  1880,  as  it 
is  found  to  represent  the  earth's  true  figure  in  this  continent 
somewhat  better  than  the  spheroid  of  Bessel. 

All  the  results  and  computations  in  the  following  pages  of 


€o. 


DIMENSIONS   OF  THE    SPHEROID. 


169 


this  volume  will  be  based  on  the  Clarke  spheroid  of  1866,  but 
it  is  well  for  the  studer\t  "to  be  acquainted  with  the  Bessel 
spheroid  also,  since  it  is  gxtensively  used  in  Europe.  The 
following  table  gives  the  complete  elements  of  the  two 
spheroids,  and  it  will  be  noted  that  the  spheroid  of  Bessel  is 
smaller  than  that  of  Clarke  and  also  less  elliptical  or  oblate. 
In  order  to  form  an  idea  of  the  precision  of  these  results  it 
may  be  noted  that  the  probable  error  of  Bessel's  quadrant  is 
498  meters  and  that  of  Clarke's  quadrant  slightly  less. 
ELEMENTS  OF  THE  SPHEROID. 


Bessel,  1841. 

Clarke,  1866. 

Semi-major  axis  a 

j  meters 
\  feet 

6  377  397 
20923597 

6  378  278 
20926062 

Semi-minor  axis  b 

j  meters 
1  feet 

6356079 
20853654 

6356654 

20855  121 

Meridian  quadrant 

in  meters 

10000856 

10001  997 

Eccentricity  <? 

0.08  1  697 

0.082271 

*' 

0.00667437 

0.00676866 

Ellipticity/ 

i 

294-98 

299-15 

The  above  values  of  the  axis  and  quadrant  of  the  Clarke 
spheroid  are  expressed  in  legal  linear  units  ^of  the  United 
States,  the  meter  being  defined  by  the  statement  that  it  is 
||^J  yards,  and  it  will  be  noticed  that  they  differ  slightly  from 
some  values  used  in  the  preceding  pages.  Clarke's  results 
were  deduced  in  feet,  and  then  transformed  into  meters  by 
the  relation  that  a  meter  contained  3.2808693  feet,  as  deter- 
mined by  his  comparisons  of  standards.  This  quantity  has 
since  been  found  to  be  too  great,  and  according  to  present 
knowledge  the  legal  ratio  of  the  United  States  is  very  closely 
the  correct  and  actual  one.  In  the  following  pages  this  legal 
ratio  will  be  used  exclusively,  namely,  one  meter  =  3.2808333 
feet;  or  the  following  logarithmic  rules  may  be  employed  to 
change  meters  into  feet: 


170  SPHEROIDAL   GEODESY.  VII. 

log  meters         +  0.5159841  =  log  feet, 
log  kilometers  +  1.7933502  =  log  miles. 

It  is  well  for  the  student  to  keep  the  first  of  these  rules  in 
the  memory,  but  should  it  be  forgotten,  it  can  be  found  by 
referring  to  the  last  page  of  the  text  of  this  book. 

Prob.  60.  Compute  the  lengths  of  one  second  of  latitude  in  feet  at 
the  pole  and  at  the  equator  of  the  Clarke  spheroid,  using  the  lengths 
of  one  degree  as  in  701  meters  and  110568  meters. 

61.  LENGTHS  OF  MERIDIAN  AND  PARALLEL  ARCS. 

The  elements  a  and  e  for  the  Clarke  spheroid,  substituted 
in  the  equations  of  Art.  57,  furnish  practical  formulas  for 
numerical  work,  To  find  an  expression  for  the  length  of  an 

arc   of   the   meridian    between    the   latitudes   L,   and   L    the 

a    \ 

general  expression  ($?)"  is  to  be   integrated  between  these 

limits;   then  representing  the   mean   latitude   \(L^  -f-  Z-a)   by 

L  and  the  amplitude  L^  —  L,  by   6,   it  can  be   put  into  the 

form 

I '  =  in  133.30$  —  32  434.25  sin#  cos2Z-{-  34.41  sin2#cos4Z, 

in  which  0  is  in  degrees  and  /  is  in  meters.  For  logarithmic 
work  this  may  be  written  in  the  more  convenient  way, 

/  =  [5.0458443]$  —  [4.5 1 10039]  sm&  cos2Z  4-  [!-5368]  sin2#  cos4Z, 

where  the  numbers  in  brackets  are  the  logarithms  of  the 
constants  in  the  first  formula.  For  instance,  to  find  the 
length  of-  the  meridian  from  latitude  45°  to  the  pole,  put 
Lt  =  45°,  Lt  =  90°,  whence  #  =  45°  and  2L  =  135°;  then 
/=  5  017  160.6  meters,  or  16460649  feet. 

1  The  length  of  one  degree  of  the  meridian  for  the  latitude 
L  is  found  by  making  8  =  i°,  and  then 

1=  if  133. 30—  [2.752859]  COS2Z  +  0.0796]  cos4Z.      (61) 

For  example,  let  L  =  37°,  then  /=  110976.3  meters,  which 
is  the  distance  on  the  meridian  from  latitude  36^°  to  latitude 


6l.  LENGTHS  OF  MERIDIAN  AND  PARALLEL  ARCS.  \J\ 

37^°.  By  dividing  this  by  60  the  length  of  one  minute 
results,  and  a  second  division  by  60  gives  the  length  of  one 
second. 

To  find  an  expression  for  the  length  of  an  arc  of  the 
parallel,  or  an  arc  of  longitude,  at  the  latitude  L,  let  the 
radius  of  this  circle  be  called  r;  then  the  circumference  is  2nr 
and  the  length  of  one  degree  is  7rr/i8o.  The  value  of  r  is 
given  by  (57),  and  by  expanding  the  radical,  and  substituting 
the  values  of  a  and  e  for  the  Clarke  spheroid,  the  length  of 
one  degree  in  meters  is 

/=  [5. 0469490]  cosL—  [1.97562]  cos3£+[l.o75i]  cos5£,    (61)' 

and  the  length  of  6  degrees  is  then  61.  For  example,  if 
L  =  89°  then  /  =  I  949.35  meters  is  the  length  of  one  degree 
of  longitude,  and  accordingly  the  distance  around  the  earth 
on  this  parallel  is  about  693  kilometers,  or  430  miles. 

A  more  expeditious  method  of  computation,  which  is 
sufficiently  accurate  for  arcs  of  the  meridian  less  than  3  or  4 
degrees,  and  strictly  correct  for  all  arcs  of  parallels,  is  by  the 
use  of  the  radius  of  curvature  for  the  given  latitude.  Thus 
if  R1  be  the  radius  of  curvature  of  the  meridian  at  latitude  Z, 
then  //-/?,  is  the  angle  in  radians,  or  /  =  Rti  is  the  length  if  6 
be  in  radians.  Accordingly 

/  =  R$  arc  i°,      when  6  is  in  degrees, 

/  =  Rfl  arc  i',      when  8  is  in  minutes,  (61)" 

/  =  R$  arc  i",     when  6  is  in  seconds. 

The  logarithms  of  R^  are  found  in  Table  IV  and  those  of 
arc  i°,  arc  i',  and  arc  \"  in  Table  VI.  For  instance,  let  it  be 
required  to  find  the  length-  in  meters  between  two  points  on 
the  same  meridian  whose  latitudes  are  39°  18'  i2//.8  and  38° 
04'  15". 2.  The  mean  latitude  L  is  38°  41'  13". 8,  and  for 
this,  by  interpolation  in  the  table,  \ogR1  is  found  to  be 
6.8034789.  The  amplitude  9  is  4437^.6  and  log#  is 
3.6461482,  while  log  arc  l"  is  6.6855749.  The  sum  of  these 


172 


SPHEROIDAL   GEODESY. 


gives  5.1352020,  whose  corresponding  number  is  136522.6, 
which  is  the  distance  in  meters  on  the  meridian  between  the 
two  given  latitudes;  if  the  result  is  desired  in  feet  the  addi- 
tion of  the  constant  (Art.  60)  gives  5.6511861,  which  is  the 
logarithm  of  447  905  feet. 

The  same  process  applies  to  obtaining  the  length  of  an  arc 
of  longitude,  the  radius  r  being  used  instead  of  R^  For 
many  rough  computations  the  values  of  the  lengths  of  arcs 
given  in  Tables  II  and  III  will  enable  numerical  work  to  be 
done  without  using  the  above  formulas. 

Prob.  6 1.  Compute  the  length  of  the  meridian  arc  from  latitude 
45°  to  the  equator,  and  also  the  length  from  the  pole  to  the  equator. 

62.  NORMAL  SECTIONS  AND  GEODESIC  LINES. 

At  any  point  on  the  spheroid  let  a  tangent  plane  be  drawn 
and  perpendicular  to  this  plane  let  a  line  be  drawn  through 
the  point;  this  line  is  called  the  normal  and  any  plane  pass- 
ing through  it  cuts  from  the 
spheroid  a  normal  section.  Of 
these  the  meridian  section  is  the 
most  important,  and  next  is  the 
X  prime  vertical  section,  which  is  the 
E  normal  section  cut  by  a  plane 
perpendicular  to  the  meridian. 
These  two  sections  are  called  prin- 
cipal normal  sections  because  the 
properties  of  all  other  normal  sec- 
tions can  be  derived  from  them.  The  figure  shows  a  point 
A  on  the  spheroid,  NS  being  the  meridian  and  WE  the  prime 
vertical  section,  while  LL  is  the  parallel  of  latitude  and  BB 
an  oblique  normal  section  through  the  point  A. 

An  expression  for  the  radius  of  curvature  of  the  meridian 
is  given  by  (57)'.  Taking  the  logarithm  of  both  members 
and  inserting  the  values  of  a  and  e*  for  the  Clarke  spheroid, 
it  can  be  put  into  the  form 


N 


*• 


62. 


NORMAL  SECTIONS  AND   GEODESIC   LINES. 


173 


^^=6.8039641  —  [3.82884]  cos2Z+[o.758]cos4£,     (62) 

where  the  numbers  in  brackets  are  logarithms  to  be  added  to 
the  logarithms  of  cos2/,  arid  0054^.  A  few  of  the  values  of 
log^?,  are  given  in  Table  IV  at  the  end  of  this  volume. 

The  radius  of  curvature  of  the  prime  vertical  normal  sec- 
tion at  its  intersection  with  the  meridian  is  the  length  of  the 
normal  AN  from  the  point  A  to  the  intersection  with  the 


minor  axis.      It  is  hence  equal  to  AD/cosL,  where  AD  is  the 
radius  of  the  parallel  given  by  (57).     Accordingly 

R,  =  a(i  -  e*  sin'Z)"* 

is  the  radius  of  curvature  of  the  prime-vertical  normal  section. 
Developing  this,  there  may  be  deduced 
log R^=6. 8054402  —  [3.35172]  cos2 £-{-[0.281]  cos4£,    (62)' 

from  which  log./?,  may  be  computed  for  any  given  value  of  L. 
A  few  of  these  values  will  be  found  in  Table  IV. 

The  radius  of  curvature  of  any  other  normal  section  at  the 
point  A  is  intermediate  in  value  between  Rl  and  R^.  If  Z  be 
the  azimuth  of  any  normal  section  at  the  point  A,  its  radius 
of  curvature  at  that  point,  as  shown  in  works  on  the  differen- 
tial calculus,  is  given  by 

~D  E>  D        ' 


which,  for  numerical  work,  is  better  written 


cos'Z 


174  SPHEROIDAL   GEODESY.  VIII 

For  example,  let  it  be  required  to  find  the  radius  of  curvatuie 
of  a  normal  section  at  a  point  A  in  latitude  39°,  its  azimuth 
being  45°.  Taking  the  logarithms  of  R,  and  R^  from 
Table  IV  and  performing  the  operations  there  is  found 
logTc  =  6.8051230,  whence  ^  =  6384443  meters.  This 
formula  is  useful  in  reducing  base  measurements  to  ocean 
level  (Art.  31). 

When  an  instrument  is  leveled  at  a  station  A  and  pointed 
to  a  second  station  By  points  set  out  in  the  line  of  sight  fall  in 
the  curve  AaB  which  is  cut  from  the 
spheroid  by  the  normal  section  at  A.  When 
the  instrument  is  leveled  at  B  and  pointed 
at  A,  the  curve  BbA  will  result,  this  being 
cut  from  the  spheroid  by  the  normal  section 
at  B.  These  two  curves  differ  very  slightly 
in  azimuth;  for  a  line  100  miles  long  the  difference  cannot 
exceed  o".  I,  so  that  it  is  of  slight  importance  in  common 
geodetic  triangulation.  These  normal  sections  are  plane 
curves. 

The  alignment  curve  between  two  stations  A  and  B  is  a 
curve  traced  by  starting  at  A,  setting  out  a  point  in  the 
direction  of  B,  then  moving  the  instrument  to  that  point, 
backsighting  on  A,  setting  a  second  point  in  the  direction  of 
B,  and  so  on.  The  broken  line  AcB  in  the  figure  represents 
this  curve,  which  is  one  of  double  curvature;  at  any  point  c 
the  vertical  tangent  plane  to  the  curve  passes  through  both 
A  and  B.  The  alignment  curve  is,  of  course,  a  shorter  path 
between  A  and  B  than  that  on  either  of  the  normal  sections. 
The  shortest  line  between  two  stations  on  the  spheroid  is 
called  a  geodesic  line,  or  simply  a  geodesic.  It  is  not  shown 
in  the  above  figure,  but  may  be  closely  represented  by  a 
curve  of  less  curvature  than  AcB  and  crossing  it  near  c\  it  is, 
like  the  alignment  curve,  a  line  of  double  curvature.  The 
geodesic  has  the  property  that  the  plane  containing  any 
element  of  the  curve  is  normal  to  the  spheroid  at  that 


63.  TRIANGLES   AND   AREAS.  175 

element.  The  differential  equation  of  the  geodesic  can  be 
deduced  and  its  properties  be  studied,  but  this  is  not 
expedient  or  necessary  in^n  elementary  book  of  this  kind, 
particularly  as  the"  line  is  of  no  importance  in  the  practical 
operations  of  geodesy. 

On  a  sphere  the  two  normal  sections  and  the  alignment 
and  geodesic  curves  between  A  and  B  coincide  in  an  arc  of  a 
great  circle.  On  a  spheroid  they  also  coincide  when  the  two 
stations  are  on  the  same  meridian;  but  in  other  cases  they 
are  separate  and  distinct.  For  any  two  intervisible  points 
on  the  earth's  surface,  however,  they  do  not  appreciably 
differ  in  length,  and  it  is  only  in  the  case  of  the  longest  lines 
that  a  difference  in  their  azimuths  can  be  detected. 

Prob.  62.  A  base  line  8046.74  meters  long  has  an  azimuth  of  about 
60°  and  its  elevation  above  ocean  level  is  1609.35  meters.  What  is 
the  length  of  the  base  reduced  to  ocean  level  ? 

63.  TRIANGLES  AND  AREAS. 

A  triangle  on  the  surface  of  a  spheroid  has  the  sum  of  its 
three  angles  greater  than  two  right  angles.  An  exact  expres- 
sion for  this  spheroidal  excess  might  be  esfablished,  but,  since 
only  triangle  sides  between  intervisible  points  can  be  used  in 
geodesy,  it  is  always  sufficiently  accurate  to  regard  these 
points  as  lying  on  the  surface  of  a  sphere  osculatory  to  the 
spheroid  at  the  middle  point  of  the  triangle.  The  radius  of 
this  osculatory  sphere  is  i/T^A:, ,  where  R^  and  R^  are  the 
radii  of  curvature  of  the  two  principal  normal  sections 
through  the  point.  Accordingly  the  formula  (55)  becomes 

Excess  in  seconds  =  206  265  area/^,^2 ,  (63) 

in  which  logT?,  and  log^3  may  be  computed  from  (62)  and 
(62)'  or  be  taken  directly  from  Table  IV,  while  the  logarithm 
of  206  265  is  found  in  Table  VI. 

As  this  computation  is  one  that   is  frequently  required  the 


CIVIL  ENGINEERING 

U.oiC 
ttSMUTMN  UPiSI 


176  SPHEROIDAL   GEODESY.  VII. 

quantity  206  26$/RlR.t  may  be  called  2m  and  the  logarithms 
of  values  of  m  for  different  latitudes  be  tabulated  as  is  done 
in  the  last  column  of  Table  IV.  Thus  the  practical  formula 
for  the  computation  of  the  excess  in  seconds  is 

Excess  =  m  .  2  Area  =  m  .  ab  sin £7,  (63)' 

in  which  ab  sin£7  is  double  the  area  of  the  triangle,  a  and  b 
being  two  sides  and  C  their  included  angle;  here  the  area 
must  be  in  square  meters,  or  a  and  b  must  be  in  meters.  For 
instance,  let  it  be  required  to  compute  the  spherical  excess 
for  a  triangle  whose  area  is  197  square  kilometers,  the  latitude 
of  its  middle  point  being  37-J-0.  From  Table  IV  the  logarithm 
of  m  for  the  given  latitude  is  taken  and  this  added  to  the 
logarithm  of  double  the  area  in  square  meters  gives  0.00026 
as  the  logarithm  of  the  excess  in  seconds,  whence  excess 
=  i".ooi,  which  differs  by  only  o".ooi  from  the  value  given 
by  the  rough  rule  of  Art.  55. 

Among  the  many  interesting  questions  relating  to  the 
spheroid  is  that  of  the  areas  of  zones  and  the  areas  of  trape- 
zoids  bounded  by  meridians  and  parallels.  The  differential 
expression  for  the  area  of  a  zone  is  2nrRldL,  where  r  is  the 
radius  of  the  parallel  and  Rl  the  radius  of  curvature  of  the 
meridian  at  the  latitude  L.  The  values  of  r  and  Rl  are 
given  by  (57)  and  (57)',  and  thus  is  found 

A  =  27ta*(i  —  V)  /  (i  —  /  sin2Z)~2  cosZ  dLt 

which,  when  integrated  between  the  limits  Ll  and  Za ,  gives 
the  area  of  the  zone  between  those  latitudes.  The  integral 
contains  a  hyperbolic  or  logarithmic  function  and  hence  is 
rather  tedious  in  computation,  but  tables  have  been  made 
giving  its  values.  Among  the  best  of  these  is  Woodward's 
Geographical  Tables,  published  by  the  Smithsonian  Institu- 
tion, where  the  areas  of  trapezoids  bounded  by  meridians 
and  parallels  are  given  in  square  miles. 

Prob.  63.  Prove  that  the  entire  surface  of  the  spheroid  is  expressed 


64.,  LATITUDES,    LONGITUDES,   AND   AZIMUTHS.  177 

by  27tai(  i  H  —      '-  loge  —     —  j,  and  show  that  this  reduces  to  471-0* 
for  the  sphere  whose  radius  is*tf. 

64.  LATITUDES,  LONGITUDES,  AND  AZIMUTHS. 

The  formulas  established  in  Art.  56  for  the  spherical 
triangle  may  be  adapted  to  any  practical  case  arising  on  the 
spheroid  with  any  required  degree 
of  precision.  The  problem  to  be 
solved  is  as  follows:  given  the  lati- 
tude and  longitude  of  a  point  A,  the 
azimuth  of  AB  and  its  length,  to  find 
the  latitude  and  longitude  of  B  and 
the  azimuth  of  BA.  The  notation 
will  be  the  same  as  that  in  Art.  56, 
the  given  latitude  and  longitude 
being  designated  by  L  and  M,  the 
given  distance  and  azimuth  by  /  and 
Z,  while  the  required  quantities  are 
L',  M',  and  Z  '.  Let  6L  be  the 
difference  in  latitude  L'  —  Z,  and  dM  be  the  difference  in 
longitude  M'  •-  M\  also  let  dZ  be  the  angle  by  which  the 
meridian  at  B  deviates  from  parallelism  to  that  at  A,  so  that 
dZ  =.  Z'  -  Z—  180°.  Then  when  dL,  dM,  and  6Z  are 
known,  the  required  quantities  will  be  given  by 


}-dZ.      (64) 

The  problem  now  is  to  find  formulas  for  computing  dL,  dM, 
and  dZ.  The  solution  here  given  will  be  sufficient  to  furnish 
the  results  correctly  to  thousandths  of  seconds  for  all  cases 
when  the  length  /  does  not  exceed  about  20  kilometers  or 
12  miles. 

Resuming  formula  (56),  and  writing  L  -f-  dL  in  place  of  L, 
it  becomes,  after  developing  the  first  member  and  dividing 
by  cosZ,, 


i;8  SPHEROIDAL   GEODESY.  VII. 

—   sm6L  =  sinS  cosZ-f-  (cosdL  —  cosS)  tanL. 

Now  as  both  6L  and  5  are  small  arcs  their  sines  may  be 
taken  as  equal  to  the  arcs  themselves;  and  also  cosS  — 
I  —  £S2  and  similarly  for  cosSL.  Accordingly  the  equation 
reduces  to 

-  dL  =  ScosZ-f  £Sa  tanZ,  —  £(££)'  tan£. 

Here  the  first  term  of  the  second  member  is  an  approximate 
value  of  6L,  the  second  term  being  small  since  it  contains  S\ 
Accordingly,  dL  in  the  third  term  may  be  replaced  by 
»S  cosZ.  Further,  since  6L  and  5  are  in  radians  the  value  of 
5  is  l/R,  if  R  be  the  radius  of  curvature  at  the  locality. 
Accordingly  the  equation  becomes 

/  cosZ       r  sin'Z  tanZ 
-  6L  =  — £-  — ^          . 

The  next  question  to  be  considered  is  regarding  the  value 
of  R.  With  regard  to  the  first  term,  which  is  the  controlling 
one,  it  plainly  should  be  the  radius  of  curvature  of  the 
meridian  passing  through  the  middle  of  the  arc,  but  as  the 
latitude  of  that  point  is  not  known  it  is  to  be  taken  as  that 
of  the  meridian  at  A.  With  regard  to  the  second  term  it  is 
not  important,  since  its  value  is  small,  what  radius  should  be 
taken,  and  it  is  customary  to  take  that  of  the  osculatory 
sphere  at  A.  Now  let  7^  and  R^  be  the  radii  of  curvature  of 
the  principal  normal  sections  at  Ay  the  values  of  these  being 
as  given  in  Art.  62 ;  then  the  equation  becomes 

/  cosZ       r  sin'Z  tanZ 

—RT        ~^R,      ' 

This  value  is  a  close  approximation,  but  it  can  be  rendered 
clpser  by  adding  a  term  to  reduce  it  to  the  radius  of  curva- 
ture of  the  meridian  at  the  middle  point  of  the  line  /.  This 

r>  r> 

term    will    be    6L    !  p -,    where   Rm   denotes   that    radius. 

•MB 

Introducing  the  general  values  of  the  radius  from  (57)'  for 


64.  LATITUDES,   LONGITUDES,   AND   AZIMUTHS.  179 


the  latitudes  L  and  Lm  ,  replacing  Lm  by  L  —  %<5L,  developing, 
and  neglecting  terms  containing  powers  of  e  higher  than  the 
square,  the  additional  term  is  found  to  be 


V 

i  -  e*  sin1/:)1 

and  accordingly  the  final  formula  for  the  difference  of  latitude 
is 

-  6L  =  /  cosZ.  B  +  /'  sin'Z  .  C  +  tf  .  D,       (64)' 

in  which  h  denotes  the  value  of  SL  as  found  from  the  first 
and  second  terms,  and  in  which  the  letters  B,  C,  and  D  are 
factors  depending  only  on  the  dimensions  of  the  spheroid' 
and  on  the  given  latitude  L.  In  order  that  dL  may  be  found 
in  seconds  the  above  expressions  for  the  constants  are  to  be 
multiplied  by  the  number  of  seconds  in  a  radian,  and  thus 

206  265  206265  tan£  }^a  sinZ,  rosZ, 

•B  =        TS       >    C  =          _  ,  .    .  -  ,   D  = 


206265(1  -e's'm'L)*' 

are  the  final  factors  which  can  be  computed  and  tabulated  for 
different  values  of  the  latitude  L. 

In  order  to  find  the  difference  of  longitude  between  A  and 
/?,  formula  (56)'  may  be  used,  Mf  —  M  being  replaced  by 
<$M.  Since  this  is  small  the  sine  may  be  replaced  by  the 
arc,  giving 

6M  =  Ss 


Here,  as  before,  5  may  be  replaced  by  l/R  and  the  value  of 
the  radius  should  be  that  of  the  prime-vertical  normal  section 
through  B.  Introducing  this,  and  reducing  from  radians  to 
seconds,  it  becomes 

/sinZ.  A'  206 

A    =  -- 


in  which  A'  is  to  be  used  for  the  latitude  L  '. 

To    find   the   difference  in   azimuth  dZ,   formula  (56)"    is 


180  SPHEROIDAL   GEODESY.  VII. 

resumed,  and  replacing  Z'  —  Z  by  180°  -j-  $2,  and  M'  —  M 
by  <5M,  it  becomes 

—  tan^tfZ  =  ta.n%6M  sin£(Z,  -\-  L')/c.os%(Lf  —  L). 

Also,  since  the  differences  of  azimuth  and  longitude  are  very 
small,  their  tangents  are  proportional  to  the  number  of 
seconds  in  their  arcs,  and 

is  final  formula  for  computing  the  difference  in  azimuth. 

The  above  formulas  were  derived  by  Hilgard  in  1846,  and 
together  with  values  of  the  logarithms  of  the  factors  Af,  B, 
Cy  and  D,  will  be  found  in  Appendix  No.  7  of  the  Coast  and 
Geodetic  Survey  Report  for  1884;  an  abridgment  of  those 
tablfes  is  given  in  Table  V  at  the  end  of  this  book,  the  proper 
change  being  made  for  the  fact  that  the  ratio  of  meter  of  that 
Appendix  to  the  meter  of  this  book  is  i.ooooii.  By  the 
help  of  these  the  computations  may  be  expeditiously  made, 
as  will  be  illustrated  in  Art.  66.  When  using  the  formulas 
in  connection  with  these  tables  it  should  be  remembered  that 
the  distance  /  must  be  taken  in  meters. 

As  a  simple  example  of  one  application  of  the  formula  for 
dL  let  it  be  required  to  find  the  number  of  seconds  in  a 
meridian  arc  whose  length  is  1 1  076.4  meters  and  whose 
southern  end  has  the  latitude  26°.  Here  Z  =  180°,  cosZ 
=  —  I,  whence  —  dL  —  —  l-B  +  (IB)*D,  and  by  the  use  of 
Table  V  there  is  found  +  dL  =  360". 953  —  o".OQ2,  so  that 
the  latitude  of  the  north  end  of  the  arc  is  26°  10'  oo".95 I. 

Prob.  64.  Given  L  =  42°,  M  =  80°,  /  =  1000  meters,  and  Z  =  90° 
for  the  point^.  Compute  L',  M ',  and  Z'  for  the  point  B. 


65.  THE   COORDINATE   SYSTEM.  l8l 


I 

4 


CHAPTER    VIII.  v 

GEODETIC    COORDINATES    AND    PROJECTIONS. 

65.  THE  COORDINATE  SYSTEM. 

The  system  of  coordinates  used  in  geodesy  is  generally  the 
angular  one  employed  in  geography,  latitudes  being  estimated 
north  and  south  from  the  equator  and  longitudes  east  and 
west  from  the  meridian  of  Greenwich.  In  North  America 
both  latitudes  and  longitudes  are  taken  as  positive  and  the 
signs  of  the  coordinates  of  a  point  are  hence  the  same  as  in 
the  linear  system  of  Art.  I.  Thus,  if  a  point  is  determined 
to  have  the  latitude  40°  19'  04" '.237  and  the  longitude 
85°  07'  3 5 ".026,  it  can  be  at  once  located  roughly  on  a  small- 
scale  map  or  be  precisely  located  on  a  large-scale  map  upon 
which  the  meridians  and  parallels  are  accurately  drawn  in  a 
certain  system  of  map  projection. 

It  is  well  to  keep  in  mind  the  approximate  rules  of  Art.  53 
regarding  the  lengths  of  one  degree,  one  minute,  and  one 
second  of  latitude.  One  second  of  latitude  being  about  31 
meters  or  101  feet,  one-tenth  of  a  second  is  about  3  meters 
or  10  feet,  one-hundredth  of  a  second  is  0,3  meters  or  I  foot, 
and  one-thousandth  of  a  second  is  0.03  meters  or  o.  I  feet. 
Precise  geodetic  work  should  hence  carry  the  latitudes  to 
thousandths  of  a  second  of  angle  in  order  to  secure  a  precision 
comparable  with  that  of  precise  plane  triangulation. 

A  second  of  longitude  is  nearly  the  same  as  that  of  latitude 
on  the  equator,  but  at  any  other  place  it  is  smaller,  a  rough 
rule  being  that  it  is  equal  to  a  second  of  latitude  multiplied 


182  GEODETIC   COORDINATES   AND   PROJECTIONS.         VIII. 

by  the  cosine  of  the  latitude.  Thus  at  latitude  40^°,  since 
cos4c£0  is  0.774,  the  length  of  one  second  on  the  parallel  is 
about  78  feet. 

The  formulas  of  Art.  61  furnish  expressions  by  which  the 
lengths  of  one  degree,  one  minute,  and  one  second  of  both 
latitude  and  longitude  can  be  computed  for  any  given  lati- 
tude L,  and  values  of  'some  of  these  will  be  found  in  Tables 
II  and  III,  These  are  sometimes  of  service  in  changing 
angular  differences  of  latitude  and  longitude  between  the 
stations  of  a  secondary  triangulation  into  linear  differences, 
but  a  more  extended  table  is  necessary  in  order  to  make  such 
computations  with  rapidity. 

The  sketch  below  gives  a  representation  of  the  coordinate 
system  of  geodesy,  the  meridians  and  parallels  being  roughly 

drawn  on  the  polyconic  projec- 
-40°50  tion  method  which  is  explained 
in  Art.  69.  Station  Pis  located 
at  latitude  40°  45'  and  at  longi- 
tude 86°  43',  while  station  P'  is 
located  at  latitude  40°  36'  and 
longitude  86°  58'.  The  straight 
87lo  87°oo'  86°so'  86V  86b3o'  line,  or  geodesic,  joining  the 
points  is  very  slightly  curved  in  the  projection.  The  azimuth 
of  PP'  is  about  53°  43',  this  being  measured  from  the  south 
around  toward  the  west.  The  azimuth  of  P'Pis  about  233° 
33',  this  being  also  measured  from  the  south  around  through 
the  west  and  north.  Owing  to  the  convergence  of  the 
meridians  that  pass  through  P  and  '  P',  the  back  azimuth  of 
P'P  differs  by  10'  from  the  azimuth  of  PP'  plus  180  degrees. 
This  coordinate  system  is  not  a  convenient  one  for  the  use 
of  local  surveyors,  but  in  an  area  of  considerable  extent  it  is 
a  necessary  one  for  the  location  of  points  in  their  relative 
positions  on  the  spheroid.  For  an  area  of  moderate  size  it 
may  be  modified  in  many  ways,  one  of  these  being  the  well- 
known  system  of  the  public  land  surveys  of  the  United 


66.  LMZ  COMPUTATIONS.  183 

States,  while  another  andjnore  satisfactory  system  is  that  of 
linear  rectangular  spherical  coordinates  which  is  described 
in  Art.  70. 

It  is  well  to  note  again  that  the  latitudes  and  longitudes 
used  in  geodetic  work  do  not  generally  agree  with  the  lati- 
tudes and  longitudes  obtained  by  astronomical  observations. 
Thus,  if  these  coordinates  be  found  astronomically  for  an 
initial  station  P  together  with  the  azimuth  of  PP',  and  if  the 
distance  PP'  be  directly  measured  or  be  found  by  computa- 
tion from  a  measured  base,  then  the  latitude  and  longitude 
of  P'  and  the  azimuth  of  P'P  can  be  computed  by  the 
methods  of  the  last  Article,  and  these  computed  values  are 
called  geodetic  ones.  If  further  astronomical  observations  be 
made  at  P' ,  the  results  will  generally  differ  from  the  geodetic 
ones  owing  to  the  plumb-line  deflection  at  P' '.  That  is  to 
say,  the  Clarke  spheroid  passed  through  P  and  oriented  by 
the  astronomical  work  done  there,  has  a  tangent  plane  at  P' 
which  is  not  parallel  to  the  astronomical  horizon  at  that  point 
(Art.  59).  The  differences  of  latitude  and  longitude  as  found 
by  geodetic  triangulation  are,  in  fact,  always  far  more  precise 
than  those  derived  from  astronomical  observations,  and  it  is 
only  by  the  field  operations  of  geodesy  that  coordinates  of 
stations  can  be  found  so  as  to  form  a  reliable  basis  for  accu- 
rate surveys. 

Prob.  65.  From  the  above  data  for  stations  P  and  P'  determine 
roughly,  with  the  help  of  Tables  II  and  III,  the  length  of  the  line 
PP'  in  miles. 

66.  LMZ  COMPUTATIONS. 

When  the  latitude  L  and  longitude  M  of  a  station  P  are 
given,  together  with  the  distance  /  and  azimuth  Z  from  it  to 
a  second  station  P',  the  latitude  L'  and  longitude  M'9 
together  with  the  back  azimuth  Z' ,  can  be  computed.  The 
formulas  of  Art.  64  will  determine  the  latitudes  and  longi- 
tudes correctly  to  thousandths  of  a  second  when  the  length 


184  GEODETIC   COORDINATES   AND   PROJECTIONS.        VIII. 

of  the  line  does  not  exceed  about  20  kilometers  or  12  miles, 
and  correctly  to  hundredths  of  a  second  for  much  longer  dis- 
tances. These  formulas  will  now  be  exemplified. 

Let  the  given  station  be  one  called  Bake  Oven,  whose 
known  latitude  is  40°  44'  54".  109  and  whose  known  longitude 
is  75°  44'  02". 222.  Let  the  distance  and  azimuth  to  a 
second  station  called  Packer  Spire  be  33  932.55  meters  and 
297°  36'  49". 42.  In  the  form  below  these  data  are  seen  in 
italic  type,  together  with  the  logarithm  of  /  in  two  places; 
then  log  s'mZ  and  log  cosZ  are  found  and  put  in  their  places, 
while  the  logarithms  of  /a  and  s'm'Z  are  found  by  doubling 
those  of  /  and  sinZ,  and  the  logarithms  of  the  factors  B,  C, 
and  D  are' taken  from  Table  V.  By  addition  the  logarithm 
of  IcosZ-B,  or//,  is  found,  and  its  double  is  the  logarithm 
of  tf '.  Then  the  logarithms  of  the  second  and  third  terms  are 
found,  and  thus  the  final  value  of  —  dL  is  5  1 i".8,  from  which 
the  latitude  of  Packer  Spire  at  once  results,  as  also  the  mean 
latitude  %(L  -f-  L').  The  longitude  computation  is  now  made 
as  indicated  by  the  formula,  the  factor^'  being  taken  for  the 
latitude  L '.  Lastly,  6Z  is  computed,  and  the  back  azimuth 
from  Packer  Spire  to  Bake  Oven  is  determined  by  the  rule 
Z'  =  Z  +  180°  +  dZ. 

A  second  or  check  computation  should  always  be  made 
whenever  there  is  another  station  that  furnishes  sufficient 
data.  For  this  case  the  latitude  of  a  station  called  Smith's 
Gap  is  40°  49'  2 1 ".787  and  its  longitude  75°  25'  21  ".906, 
while  the  distance  and  azimuth  from  it  to  Packer  Spire  are 
24  332.28  meters  and  351°  il'  08". 84.  Inserting  these  data 
in  another  form,  and  carrying  out  the  computations  in  the 
same  manner,  the  value  of  L'  will  be  found  to  agree  within 
o".oo6,  or  0.6  feet,  with  that  of  the  first  computation,  while 
the  value  of  Mf  will  be  found  to  agree  within  o".oo2.  These 
discrepancies  are  due  to  the  fact  that  both  lines  exceed  20 
kilometers  in  length.  The  back  azimuth  from  Packer  Spire 
to  Smith's  Gap  is  found  to  be  171°  12'  52". 29,  and  the 


66. 


LMZ  COMPUTATIONS. 
FORM   FOR   LMZ  COMPUTATION. 


i8S 


PACKER  SPIRE  computed  from  BAKE  OVEN. 
**                                    ABAKE  OVEN 

-  8L    -  tcosZ.  B'+l1  sin'Z.  C+  tf.D                   ^""^X^ 
-j-  SAf  =  tsinZ.A'/cosL'                                                            ^^^^ 
—  dZ  =  d  M  sm\(L  -$-  L')                                                         PACKER  SPIRED 

Z               Bake  Oven 
SZ 

180° 

to  Packer  Spire             297°  36'  49"  .42 

+  13    53  -59 
180 

Z'                Packer  Spire  to  Bake  Oven             117°  50'  43".  01 

L      40*  44'  34".  1  09                 Bake  Oven                M      7  5°  44'  02"  .222 
SL         —08    31   .859           /  =  3}  932.55  meters        dM       —21    18  .904 

L'        40°  36'  22".  250 

Packer  Spire               M'      75°  22'  43".3i8 

/               4,3306166 
cosZ          1.6660576 
B             2.5107900 

/*             9.06123 
sin'Z          1.89496 
C             9.33974 

h*         5.4149 

D         8.3882 

h              2.7074642 
ist  term  +  509".  8755 
2d  and  3d  terms      i  .9831 

0.29593 
2d  term       i".9767 

8.8031 
3d  term  o".oo64 

/            4.3306166 
sinZ          1.9474790 
A'           2.5090982 
cosL'(a.f)  0.1196432 

dM         3.1068370 
s\n\(L+L')  1.8141128 

-  SL               +  5ii  .859 

\$L              _  04'  I5".93 
l(L  +  Lf)    40°  40'  38".  18 

3.1068370 
+  dM  -  I278".904 

2.9209498 
-dZ        -  833".59 

difference  between  this  and  the  back  azimuth  to  Bake  Oven 
is  53°  227  09". 28,  which  furnishes  a  final  check  on  the  work, 
as  this  is  the  value  of  the  spherical  angle  at  Packer  Spire. 

The  best  way  to  carry  on  these  two  computations  is  to 
enter  the  data  in  both,  find  log  sinZ  and  log  cosZ  for  both 
tt  the  same  time,  take  out  the  factors  B,  C,  and  D  for  both, 


186  GEODETIC   COORDINATES   AND    PROJECTIONS.        VIII. 

and  thus  finish  the  computation  of  6L  in  both  at  the  same 
time.  If  these  values  agree,  as  they  should  unless  the  lines 
are  too  long,  the  two  computations  for  SM  may  be  made,  and 
lastly  the  two  for  dZ.  The  signs  of  6L  and  6M  can  in  all 
cases  be  found  from  the  signs  of  cos^  and  sinZ,  but  it  will 
be  just  as  well  for  the  student  to  determine  them  from  the 
figure  that  should  always  be  drawn  at  the  top  of  each  com- 
putation sheet. 

The  above  formulas  may  be  used  in  finding  coordinates  to 
tenth  of  seconds  or  to  single  seconds  for  primary  lines  of 
almost  any  length,  but  when  these  are  required  to  thousandths 
of  seconds  additional  terms  are  needed.  These  terms  and 
the  form  for  computation  will  not  be  presented  in  this  ele-x 
mentary  book,  but  they  may  be  found  in  the  paper  of  the 
Coast  and  Geodetic  Survey  cited  in  Art.  64. 

Prob.  66.  Using  the  above  latitudes  and  longitudes  of  the  stations 
Bake  Oven  and  Smith  Gap,  and  also  the  data  that  the  distance  and 
azimuth  from  Bake  Oven  to  Topton  are  30433.63  meters  and 
351°  48'  49".  n,  and  from  Smith  Gap  to  Topton  are  44  239.59  meters 
and  29°  54'  i7".84,  make  the  two  LMZ  computations  for  Topton, 
and  check  the  back  azimuths  by  comparison  with  the  spherical  angle 
at  Topton,  whose  value  is  37°  53'  i9".oi. 

67.  THE  INVERSE  LMZ  PROBLEM. 

When  the  latitudes  and  longitudes  of  two  stations  are 
given,  it  is  possible,  if  they  are  not  too  far  apart,  to  compute 
the  length  of  line  joining  them  and  the  front  and  back 
azimuths  of  that  line.  This  is  readily  done  in  a  plane  system 
of  coordinates,  as  illustrated  in  Art.  24,  but  in  a  geodetic 
system  it  is  more  difficult.  This  is  called  the  inverse  LMZ 
problem,  and  it  will  now  be  shown  how  the  formulas  of 
Art,  64  are  applied  to  its  solution. 

Since  the  latitudes  L  and  L'  are  given,  as  also  the  longi- 
tudes M  and  M',  the  values  of  6L  and  dM  are  known.  Then 
formulas  (64)'  and  (64)"  may  be  written  in  the  form 


7^  ^    A  *-/' 

67.  THE   INVERSE   ZJ/Z  PROBLEM.  187 

-  tfZ    =  (/  cosZ)^  +  (I  sinZyC  +  (6Df  .  D,     (67) 
+  dM  =  (I  s&Z)Ay/cosL',  (67)' 

in  which  /and  Z  are  two  Unknown  quantities  to  be  found. 
From  (67)'  the  value  of  (/sinZ)  at  once  results  and  this 
inserted  in  (67)  gives  the  value  of  (/cosZ);  then,  by  dividing 
the  former  by  the  'latter  tanZ  is  found  and  hence  Z.  Also 
dividing  (/sinZ)  by  sinZ  the  value  of  /  results.  Lastly  6Z  is 
computed  by  (64)'"  and  Zf  by  (64).  The  form  used  in 
Art.  66  may  be  advantageously  employed  in  making  the 
computations,  as  will  now  be  exemplified. 

Let  the  latitudes  and  longitudes  of  the  stations  Smith  Gap 
and  Bake  Oven  be  given  as  stated  in  the  last  Article,  and  let 
it  be  required  to  compute  /,  Z,  and  Z'.  These,  with  the 
resulting  values  of  SL  and  SM,  are  first  inserted  in  the  form 
as  seen  in  italic  type.  From  Table  V  the  factors  A',  B,  C, 
and  D  are  taken.  The  logarithm  of  dM  is  found,  then  those 
of  cosZ'  and  A',  and  accordingly  the  logarithm  of  (/  sinZ) 
results.  From  dL  the  logarithm  of  (dL)*  is  found,  and  that 
of  (/sinZ)2  being  also  known,  the  second  and  third  terms  of 
the  value  of  dL  are  determined,  and  finally  the  first  term 
whose  logarithm  is  then  known  and  from  which  the  logarithm 
of  (/  cosZ)  results.  Then  tanZ  is  obtained  as  expjained 
above  whence  Z  is  found;  then  the  logarithm  of/  and  its 
value  in  meters  is  determined.  Lastly,  the  computation  of 
dZ  is  made  and  the  back  azimuth  Z'  is  obtained. 

A  check  computation  for  this  case  should  also  be  made  by 
changing  the  order  of  the  stations;  thus  the  values  of  L  and 
M  may  be  taken  for  the  station  Bake  Oven  and  L'  and  M' 
for  the  station  Smith's  Gap.  If  the  lengths  of  the  lines  do 
not  exceed  20  kilometers  the  values  of  the  lengths  and 
azimuths  should  exactly  agree  with  those  of  the  first  com- 
putation. This  inverse  solution  is  often  advantageous  in  field 
work  in  determining  the  directions  between  stations  which 
are  not-connected  by  a  triangle  side. 


188  GEODETIC   COORDINATES   AND    PROJECTIONS.        VIII. 

INVERSE  LMZ  COMPUTATION. 


A 
A             SMITHS  GAP/A 

-  dL  =  (/cosZ)^  +  (/sinZ)«C  +  (<5Z)2Z>                    £               .^^\ 

+  3M=(isinZ)A'/cosL'                                                  ''^^^           Z\ 
CTBAKE  OVEN 
-  3Z  =  dM  sin£(Z  +  Z') 

Z               Smith's  Gap  to  Bake  Oven 
SZ 
180° 

Z'               Bake  Oven  to  Smith's  Gap 

72°  39'  07".24 

—     12    II     .82 

1  80 

252°  26'  55".42 

L 
$L 

L' 

40°  49'  2i".787                 Smith's  Gap                M      75°  23'  21".  god 
—  04   27  .678          /=  27535.63  meters      dM       —    18   40  .j/6 

40°  44'  54  '.109 

Bake  Oven               M'      75°  44'  02".  222 

2d 

(cosz)         3.9I436I9 
B              2.5107842 

(cok)        8'83935 
C            §.34087 

(«Z)«     48552 
D        8.3884 

h               24251461 
ist  term     —  266".  162 
and  3d  terms  +  I  .516 
-  SL      —  267".  678 

logdL              2.42762 
$(£+£')  40°  47'  07".  9 

0.18022 
2d  term     -f-  1.5143 

3  2436 
3d  term    -f  o".ooi8 

(sinz)         4.4196765 
A'             §.5090946 
cosZ'(a  .  f)  0.1205694 

dM        3.049340 
s\nl(L+L')  1.815066 

3.0493405 
+  dM  —  1  120".  316 

2.864406 
-SZ        -  73i°.82 

(/sinZ)      4.4196765                              (/sinZ)     4.4196765 
(/cosZ)     3.9143619                                 sinZ       1.9797815 

tanZ       0.5053146                                    /          4.4398950 

Prob.  67.  Make  the  inverse  LMZ  computation  for  the  above  data, 
taking  L  and  M  for  the  station  Bake  Oven  and  L'  and  M'  for  the 
station  Smith's  Gap. 


68. 


MAP  PROJECTIONS. 


I89 


68.  MAp  PROJECTIONS. 

p 

As  a  surface  of  double  curvature  cannot  be  developed  on  a 
plane  it  is  impossible  to  devise  any  method  of  representing  a 
large  area  on  a  map  without  some  distor- 
tion. The  method  of  orthographic  pro- 
jection is  perfectly  satisfactory  for  a  small 
area,  but  when  applied  to  the  whole 
earth,  or  even  to  a  large  county,  the 
features  near  the  edges  of  the  map  are 
crowded  together  so  as  to  appear  un- 
natural. For  instance,  in  the  lower  dia- 
gram of  the  figure,  which  shows  an  ortho- 
graphic projection  of  the  northern 
hemisphere  on  the  plane  of  the  equator, 
it  is  seen  that  the  distance  between 
parallels  of  latitude  near  the  outer  parts 
of  the  map  is  much  less  than  near  the  central  part;  in  the 
upper  diagram,  which  is  an  orthographic  projection  on  the 
plane  of  one  of  the  meridians,  a  similar  distortion  is  also 
observed. 

A  projection  devised  by  Flamsted  to  avoid  this  distortion 
consists  in  dividing  the  central  meridian  NS  into  parts  pro- 
portional to  the  distance  be- 
tween the  parallels,  and  through 
these  points  drawing  straight 
lines  to  represent  those  paral- 
lels. Each  parallel  is  then 
divided  into  the  same  number 
of  equal  parts,  and  the  merid- 
ians are  drawn  through  these 


rm 


\\  \  \ 


\  \  \  \ 


Q 


§ 


points  of  division.  All  the  trapezoids  between  two  given 
parallels  thus  have  the  same  area,  and  the  distortion  is  less 
than  in  the  orthographic  projection. 


GEODETIC   COORDINATES   AND   PROJECTIONS.        VIII. 


A  projection  devised  by  Bonne  is  constructed  in  a  similar 
manner  to  that  of  Flamsted,  except  that  the  parallels  are 
concentric  circles.  The  center  of  these  circles  is  in  the 
middle  meridian  and  at  a  distance  of  a  co(L  from  the  middle 
parallel  whose  latitude  is  L\  thus  in  representing  half  of  the 
hemisphere  the  radius  of  the  middle  parallel  is  equal  to  the 

y 


equatorial  radius  a,  and  the  radius  of  any  other  parallel  is 
a  ±  d,  where  d  is  its  distance  from  the  middle  parallel.  This 
method  gives  trapezoids  of  equal  area  and  an  orthographic 
projection  along  the  middle  meridian  and  parallel,  but  the 
shape  becomes  rather  awkward  where  a  large  area  is  repre- 
sented. 

The  cylindrical  map  projection  or  development  is  made 
by  projecting  the  parallels  upon  a  circumscribing  cylinder  by 
lines  drawn  from  the  center  of  the  sphere,  and  then  develop- 
ing the  cylindrical  surface.  If  QPQ  be  a  meridian  section  of 
the  spheroid  and  d  any  point  upon  it,  then  d  is  projected  at 
D  on  the  cylinder,  and  thus  the  parallel  through  d  is  projected 
upon  the  cylinder  in  a  circle  whose  diameter  is  D' D.  The 
cylinder  being  developed  on  a  plane  tangent  to  the  cylinder, 
the  circle  D' D  rolls  out  into  the  straight  line  D,D^  while  the 
equator  rolls  out  in  its  true  length  on  the  line  Q^.  If  L 
be  the  latitude  of  any  parallel  and  R  the  radius  of  the 
sphere,  the  distance  of  the  parallel  from  the  equator  on  the 
development  is  R  tanZ.  Thus  the  distances  between 
parallels  increases  toward  the  poles  and  the  poles  themselves 
cannot  be  shown  on  this  projection.  The  equator  being 
divided  into  equal  parts,  representing  degrees  of  longitude, 


68. 


MAP   PROJECTIONS. 


the  meridians  are  drawn  parallel  to  each  other,  and  accord- 
ingly one  degree  of  longitude  has  the  same  length  on  all  parts 
of  the  map.  Along  the  eqtfator  the  projection  is  orthographic, 


Di 

D2 

19 

50° 

35" 

0° 

25° 

50" 

75° 
0° 

Qi 

Q2 

!o°          12 

0°          61 

1°             ( 

6 

0°          K 

»°      11 

and  a  distance  may  be  laid  off  in  its  true  length.  At  any 
latitude  L  a  distance  /  has  the  length  /  secZ  when  laid  off 
along  a  parallel  and  the  length  /  sec2Z,  when  laid  off  along  a 
meridian.  In  the  polar  regions  the  distortion  is  so  great  that 
the  projection  is  unsatisfactory. 

Mercator's  map  projection  is  a  favorite  one  among  naviga- 
tors, since  the  course  of  a  ship  may  be  plotted  upon  it  in  a 
straight  line  as  long  as  the  ship  sails  on  the  same  true  bearing. 
It  resembles  the  cylindrical  projection  in  having  parallel 
straight  lines  for  the  meridians,  while  the  circles  of  latitude 
are  also  straight  lines  normal  to  the  meridians,  but  the  dis- 
tance on  the  projection  of  any  circle  of  latitude  from  the  equa- 
tor is  derived  from  the  condition  that  the  spiral  line,  or  loxo- 
drome,  on  which  the  ship  sails,  shall  be  a  straight  line  on  the 
projection.  Let  the  left-hand  diagram  in  the  following  figure 
represent  a  part  of  the  sphere,  QC  and  Q1C1  being  meridians 
and  CCl  a  circle  of  latitude,  while  CE1E2  is  the  loxodrome 
which  makes  the  same  angle  with  all  meridians.  Let  the 
right-hand  diagram  represent  the  Mercator  projection,  where 


GEODETIC    COORDINATES   AND    PROJECTIONS.        VIII. 


the  corresponding  meridians  are  qc  and  q^c^  while  ccl  is  the 
circle  of  latitude  and  cee^  the  straight  line  representing  the 
loxodrome.  The!  condition  of  equal  azimuths  for  the  loxo- 
drome  in  the  two  figures  is  that  the  ratio  clejccl  shall  equal 
the  ratio  C^E  /CCl  when  the  two  meridians  are  very  near 
together.  Then."^,^,  may  be  called  dy  where  y  is  the  distance 
qc  which  is  to  be  determined.  Now  ccl  is  equal  to  qql  or  to 
QQlt  and,  if  R  be  the  radius  of  the  sphere  and  L  the  latitude 
of  C,  the  value  of  ClEl  is  R.dL9  while  that  of  CC,  is  QQ^cosL. 


Q2 


Accordingly  dy  =  RdL/cosL  is  the  condition  that  the  azi- 
muths of  CEfii  and  cete^  shall  be  equal  and  that  cete%  shall  be 
a  straight  line.  The  integration  of  this  gives 


or 


=  2.3025851  R  log  tan(45 


where  the  logarithmic  tangent  is  in  the  common  system. 
Values  of  y  for  different"  values  of  L  may  be  computed  from 
this  formula,  and  the  Mercator  projection  of  the  surface  of  the 
sphere  is  then  easily  constructed. 

For  the  Mercator  projection  of  the  surface  of  a  spheroid  a 
formula  for  the  distance  qc  or  y  may  be  obtained  in  a  similar 
manner  from  the  condition  clel/ccl  =  C^EjCC^.  Here  clel  is 
dy  and  ccl  is  a.d\,  where  a  is  the  semi-major  axis  of  the 
spheroid  and  d\  is  the  difference  in  longitude  between  the  two 
meridians;  also  ClEl  is  RrdL,  where  R^  is  the  radius  of  curva- 
ture of  the  meridian  at  the  latitude  L,  and  CCt  is  r.d\  where 


68.  MAP   PROJECTIONS.  \g\b 

r  is  the  radius  of  the  circle  of  latitude.      The  values  of  r  and 
R^  are  given  in  (57)  and  (57)',  and  the  condition  reduces  to 

- 

y  ~ 


cosL(i  —  e2  sin2Z)  * 

This  expression  may  be  integrated,  and,  neglecting  terms  con- 
taining powers  of  e  higher  than  the  fourth,  the  integral  is 

y  —  2.3025851  a  log  tan  (45°  -|-  \L]  —  ae2  sinZ,  —  \ae*  sm3L, 
from  which  values  of  y  are  readily  computed. 

Mercator's  projection  is  mostly  used  on  the  ocean,  where  the 
nautical  mile  is  the  unit  of  measure.  Taking  a  and  e  for  the 
Bessel  spheroid,  and  using  1855.11  meters  as  the  length  of  a 
nautical  mile,  the  value  of  y  in  nautical  miles  is 

y  —  791  5.705  log  tan  (45°  +  JZ)  —  22.945  sinZ  —  0.05  I  sin3Z. 

For  example,  the  computed  values  of  y  at  intervals  of  5°  from 
latitude  30°  to  latitude  50°  are  as  follows  : 

£±       30°  35°  40°  45°  50° 

7=1876.9       2231.1       2607.9       30*3.7       3456.9 
4y=  354-2         376.8         405.8         443.2 

and  by  using  the  differences  in  the  last  line  a  Mercator  pro- 
jection is  readily  constructed,  the  distance  between  two  merid- 
ians 5  degrees  apart  being  exactly  300  nautical  miles.  Books 
on  navigation  give  values  of  y  for  every  degree  and  minute  of 
latitude,  as  they  are  needed  both  for  the  construction  of  charts 
and  in  nautical  computations. 

Prob.  68.  For  a  cylindrical  projection  show  that  the  distance  of 
any  parallel  of  latitude  from  the  equator  is  given  by 

y  =  a  tanZ  —  a?  sinZ(i  —  <r2  sin'Z)~*. 

Prob.  680.  Consult  a  work  on  navigation  and  ascertain  the 
meaning  of  the  table  of  meridional  parts;  also  how  it  is  used  to  find 
the  distance  from  one  port  to  another  and  the  azimuth  of  the  loxo- 
drome  connecting  them. 


GEODETIC   COORDINATES   AND    PROJECTIONS.        VIII. 


69.  THE  POLYCONIC  PROJECTION. 

The  map  projection  that  is  used  exclusively  in  geodetic 
work  is  one  in  which  each  parallel  circle  of  latitude  is  devel- 
oped on  a  conical  surface,  there  being  as  many  cones  as  there 
are  parallels.  In  the  figure  let  A  be  any  point  on  the  spheroid 


whose  latitude  is  L  and  let  r  be  the  radius  of  its  parallel  of 
latitude  whose  value  is  given  by  (57).  Let  a  tangent  cone  be 
drawn  touching  the  spheroid  at  this  circle  of  latitude;  in  the 
figure  A  T  is  one  element  of  this  cone,  its  vertex  being  at  T 
where  A  T  meets  the  polar  axis.  If  this  cone  be  developed, 
as  in  the  second  diagram,  the  parallel  of  latitude  rolls  out 
into  a  circle  ASAlt  whose  radius  is  the  same  as  TA.  In  the 
same  manner  a  tangent  cone  may  be  rolled  out  for  any  other 
circle  of  latitude,  its  radius  being  different  from  that  in  the 
first  case. 

For  large-scale  maps  the  radius  TA,  or  r',  is  so  long  that 
it  is  impracticable  to  describe  the  circle  ASA^  with  the  com- 
pass, and  hence  it  is  usually  constructed  by  finding  the 
abscissas  x  and  the  ordinates  y  with  respect  to  a  point  5  on 
the  central  meridian  TS.  Since  the  angle  A  TN  is  the  same 
as  L  the  value  of  r'  is-r/smL.  Now  if  it  be  desired  that  SA^ 
shall  correspond  to  n  degrees  of  longitude,  the  length  of  this 
parallel  arc  is  rn,  but  in  the  projection  its  length  is  r'6,  where 
6  is  the  angle  ^JS.  Therefore,  equating  these,  the  value 
of  6  is  given  by 

B  =  n  sinL.  (69) 


69. 


THE   POLYCONIC   PROJECTION. 


193 


for  n  = 

2° 

e  = 

i°  I7'o8".o7 

X   = 

170780 

y  = 

i  916 

2°  34' 


After  6  is  found  the  abscissa  and  ordinate  result  from 

x  =  r'  sin#,         y  =?  r'(i  —  cos#)  =  2r'  sin'ifl,     (69)' 

in  which  r'  has  th£  value  Stated  above.  In  numerical  work 
"t  is  preferable  to  express  r'  in  terms  of  Rt,  the  radius  of 
curvature  of  the  prime-vertical  section  at  A  (Art.  62),  whose 
value  is  the  same  aS  that  of  the  line  AN.  Since  the  triangle 
NAT  is  right-angled,  it  follows  that 

r'  =  R,  cotZ,  (69)" 

and  thus  r'  can  be  easily  computed  by  the  help  of  Table  IV. 

For  example,  at  latitude  40°  let  it  be  required  to  find  the 
values  of  x  and  y,  for  2°,  4°  and  6°  of  longitude.  From 
Table  IV  the  logarithm  of  Rt  is  6.8053115,  and  accordingly 

4°  6° 

i6".i4  3°  51'  24".2i 

\.i  475  5  1 1  996  meters 

7663  17238  meters 

and  by  these  three  points  may  be  located  on  each  side  of  the 
central  point  5  of  the  developed  circle  of  latitude. 

In  order  to  construct  a  polyconic  projection  for  an  area 
embracing  twelve  degrees  of  longitude  and  eight  degrees  of 
latitude,  with  meridians  at  . 
intervals  of  two  degrees,  five 
computations  like  the  above 
are  to  be  made.  If  the  am- 
plitude is  from  latitude  40° 
to  latitude  48°,  a  straight  line 
NS  is  drawn  for  the  central 
meridian.  At  5  a  straight  40' 
line  normal  to  NS  is  drawn, 
and  then  six  points  on  the  40°  circle  are  located  with  the  help 
of  the  above  values  of  x  and  y.  The  distances  SU,  UV, 
VW,  and  WN  are  next  laid  off  by  the  help  of  Table  II  and 
through  the  points  thus  determined  straight  lines  are  drawn 


44 


80 


u 


3V 


[94  GEODETIC    COORDINATES   AND    PROJECTIONS.        VIII. 

normal  to  NS,  and  on  each  of  these  the  values  of  x  and  y  are 
kid  off  as  computed  for  the  latitudes  42°,  44°,  46°,  and  48°, 
thus  locating  six  points  on  each  parallel.  Through  these 
points  curves  are  drawn  and  the  coordinate  system  is  com- 
pleted. 

Tables  of  the  lengths  of  arcs  of  the  meridian  and  of  values 
of  the  coordinates  x  and  y  are  indispensable  in  the  construc- 
tion of  polyconic  projections.  Extended  tables  in  meters 
may  be  found  in  Appendix  No.  6  of  the  Report  of  the  U.  S. 
Coast  and  Geodetic  Survey  for  1884,  while  similar  ones  for 
the  English  system  may  be  found  in  Woodward's  Geographi- 
cal Tables,  published  by  the  U.  S.  Smithsonian  Institution. 

The  polyconic  projection  furnishes  a  system  of  coordinates 
that  gives  an  excellent  representation  of  the  earth's  surface 
or  of  any  part  of  it.  The  parallels  and  meridians  everywhere 
intersect  at  right  angles.  Distances  are  correctly  represented 
on  the  central  meridian  and  on  all  the  parallels.  The  dis- 
tances on  meridians  near  the  borders  of  the  map  are  a  little 
too  long,  and  it  is  here  that  the  distortion  is  greatest.  On 
the  whole  this  method  gives  the  best  projection  of  angles 
with  the  lenst  possible  distortion  of  figures. 

Prob.  69.  Make  all  the  necessary  computations  and  construct  a 
polyconic  projection  on  a  scale  of  1/2  oooooo  for  the  portion  of  the 
spheroid  indicated  by  the  above  figure. 

70.  LINEAR  SPHERICAL  COORDINATES. 

For  the  purposes  of  local  surveys  the  angular  system  of 
coordinates  is  not  a  convenient  one,  as  surveyors  require  all 
distances  to  be  expressed  in  linear  measures.  The  system  of 
rectangular  linear  spherical  coordinates,  now  to  be  described, 
is  a  generalization  of  the  linear  method  given  in  Chapter  II, 
and  can  be  applied  satisfactorily  to  a  territory  of  several 
thousand  square  miles.  This  system  has  been  extensively 
used  in  Europe,  but  is  little  known  in  America,  where  precise 
detailed  surveys  of  large  areas  have  as  yet  not  been  made. 


LINEAR   SPHERICAL  COORDINATES. 


195 


Let  O  be  an  origin  of  coordinates  at  the  center  of  the  terri- 
tory to  be  covered  by  the  system,  TVS  a  central  meridian,  and 
R  the  radius  of  curvature  of  the 
spheroid  at  O,  that  is,  the  radius  of  a 
sphere  osculatory  to  the  spheroid  at 
the  origin.  Through  any  point  Pl 
let  an  arc  of  a  great  circle  of  this 
osculatory  sphere  be  drawn  normal 
to  the  meridian  TVS,  meeting  it  in 
Mr  Then  the  linear  distances  OMl 
and  MlPl  are  the  linear  rectangular 
spherical  coordinates  of  P^  and  these 
will  be  expressed  by  the  letters  Ll  and  My.  Similarly,  for 
another  point  P9  the  latitude  £2  is  the  distance  OM9,  and  the 
longitude  M^  is  the  distance  M^.  Here  latitudes  are  taken 
as  positive  when  measured  northward  of  (9,  and  longitudes  as 
positive  when  measured  westward  from  TVS. 

Through  Pl  and  P2  let  circles  be  drawn  parallel  to  the 
central  meridian  TVS,  and  let  the  angle  which  the  line  Pf* 
makes  with  the  circle  through  Pl  be  called  Z^  Here  Z,  is 
not  strictly  an  azimuth,  and  it  is  best  to  call  it  a  direction- 
angle.  Similarly,  the  direction-angle  of  P^Pl  at  P2  is 
called  Z2.  These  direction-angles  are  measured  from  the 
south  around  through  the  west,  north,  and  east  exactly  like 
geodetic  azimuths. 

Let  /  be  the  length  of  the  line  Pf^ ,  and  let  also  the  lati- 
tude Z, ,  the  longitude  M^ ,  and  the  direction-angle  Z,  be 
given;  it  is  required  to  find  the  latitude  L^ ,  the  longitude 
M9 ,  and  the  direction-angle  Z2.  The  solution  in  the  case  of 
a  plane  is 

—  /  cosZj ,     MI  =  Ml  +  /  sinZ, ,     Z,  =  Z,  +  180°, 


A  = 


and  in  the  case  of  the  sphere  the  same  expressions  will  result 
with  the  addition  of  terms  of  small  numerical  value  which 
contain  the  radius  of  curvature  R.  The  formulas  here  given 


196  GEODETIC   COORDINATES   AND    PROJECTIONS.        VIII. 

are  those  deduced  by  Soldner  in  1809;  the  demonstration, 
though  not  difficult,  will  be  omitted.  First,  let  m  and  n  be 
computed  from 

m  =  /  cosZ,  ,          n  —  I  sinZj. 

Then  the  required  quantities  are  determined  by 


mM?       mn* 
- 


206  2l 


in  which  the  terms  containing  R*  are  easily  computed  by  the 
use  of  four-place  logarithms.  Since  R*  =  R^  ,  where  Rl 
and  R^  are  the  radii  of  curvature  of  the  two  principal  normal 
sections,  the  logarithm  of  R*  is  readily  found  from  Table  IV 
at  the  end  of  this  book. 

For  example,  suppose  such  a  coordinate  system  to  be  used 
for  a  region  whose  middle  latitude  is  40°.  From  Table  IV 
the  logarithms  of  the  constants  are,  in  meters, 

log  (1/2R*)  =  18.0901,          log  (i/6R>)  =  14.6130, 
log  (206  26$/R*)  =  9.7055,     log  (206  26$/2R*)  =  9.4045- 

Now    take    /=  27516.0   metres,  M,  =  +42585.934   meters, 
L,  =  +  51  449.866   meters,    and   Z,  =  16°  47'  06".  38,   while 
the   length   of  the  line  Pf*    is  /  =  27516.0   meters.     Then 
m  =  +  26  343.669  meters,  n  =  +  7  946.133  meters,  and 

J/a  =  42  585.9344-  7  946.i33+o-233-o.o23  =+5o  532.277  meters, 
Lt  =  51  449.866  —  26  343.669  —  o.i  23  -{-0.007  =  +  25  106.081  meters, 
Za  =  16°  47'  o9".38  +  180°  -  o3".66-f  oo//.53  =  196°  47'  o6".25, 

and  thus  the  point  P^  is  completely  determined. 

One  of  the  great  advantages  of  this  system  is  that  the 
difference  of  the  front  and  back  direction-angles  of  a  line 
differs  but  little  from  180°,  and  either  may  be  used  by  a  local 


?O.  LINEAR    SPHERICAL   COORDINATES.  197 

surveyor  to  check  his  topographic  work.  With  geodetic 
azimuths,  on  the  other ,  hand,  the  orientation  of  such  local 
work  may  be  more  accurately  made  at  the  starting  station, 
but  when  checking  on  a  second  station  the  large  difference  in 
direction  is  liable  to  lead  to  confusion.  Undoubtedly  the 
system  of  linear  spherical  coordinates  must  in  time  come  into 
use  in  America,  and  by  it  or  some  similar  method  the  results 
of  the  geodetic  triangulations  can  be  made  more  generally 
available  for  use  in  precise  detailed  surveys  of  large  areas. 
For  a  fuller  account  of  the  system,  as  also  for  the  method  of 
rinding  the  distance  and  direction-angle  between  two  stations 
whose  latitudes  and  longitudes  are  given,  reference  is  made 
to  Vol.  II  of  Jordan's  Handbuch  der  Vermessungskunde. 

Prob.  70.  Given  Lv  =  -j-  50  ooo,  J/,  —  —  10  ooc,  Z,  =  +  60  ooo, 
and  J/,,  =  —  20  ooo  meters,  to  find  the  distance  from  Pl  to  P%  and 
the  direction-angles  Zl  and  Za. 


198  GEODETIC    TRIANGULATION.  IX. 


CHAPTER  IX. 
GEODETIC    TRIANGULATION. 

71.  RECONNAISSANCE. 

The  first  thing  to  be  done  in  a  reconnaissance  for  selecting 
the  stations  of  a  geodetic  triangulation  is  to  make  a  careful 
study  of  all  existing  maps.  Small-scale  sketch  maps  should 
be  prepared,  showing  the  principal  watersheds  and  mountain 
ranges  as  far  as  they  are  known,  and  these  are  to  be  taken 
into  the  field  by  the  reconnaissance  party.  Such  a  party 
consists  of  two  or  three  men  and  it  is  equipped  with  aneroid 
barometers,  prismatic  pocket  compasses,  binocular  field 
glasses,  and  photographic  cameras,  together  with  apparatus 
for  climbing  trees. 

Ascending  to  one  of  the  highest  elevations  in  the  region  a 
series  of  sketches  showing  the  visible  horizon  and  intermediate 
hill  ranges  is  made.  On  this  are  marked  the  magnetic  bear- 
ings and  the  estimated  distances  to  all  prominent  peaks  and 
gaps.  Photographs  of  the  portions  of  the  horizon  where  it 
seems  probable  that  stations  may  be  located  should  also  be 
taken,  and  the  names  of  all  mountain  ranges  and  peaks  be 
ascertained.  Then,  ascending  to  another  elevation  several 
miles  away,  a  similar  series  of  sketches  is  made,  and  after 
several  of  these  observations  the  party  obtains  a  fair  idea  of 
the  topography  of  the  country.  The  heights  of  all  the  posi- 
tions occupied  in  this  work  are  to  be  obtained  as  closely  as 
can  be  done  with  the  aneroid. 

The  results  of  these  operations  are  to  be  plotted  from  day 


71.  RECONNAISSANCE.  199 

to  day  on  the  sketch  maps,  the  visible  horizon  as  seen  at  each 
station  being  roughly  drawn,  and  the  intersection  of  these 
horizon  lines,  together  with  the  observed  heights,  will  give 
information  regarding  the  approximate  positions  to  be  selected 
for  the  primary  stations.  In  many  cases  there  are  one  or  two 
peaks  so  prominent  that  no  doubt  exists  as  to  their  avail- 
ability for  stations,  while  regarding  others  much  additional 
field  work  must  be  done  before  a  final  decision  can  be  made. 
The  intervisibility  of  adjacent  stations  must  of  course  be 
insured,  and  in  a  prairie  country  where  high  towers  are  to  be 
erected  this  requires  the  application  of  the  rules  of  Art.  37 
regarding  curvature  of  the  earth  and  refraction.  The  primary 
stations  are  to  be  so  selected  as  to  secure  the  best  triangle, 
polygon,  or  quadrilateral  nets  to  cover  the  given  area  under 
the  prescribed  conditions  of  precision  and  cost,  care  being 
taken  to  avoid  angles  less  than  30  degrees,  except  in  quadri- 
laterals (Art.  17).  As  a  general  rule  for  primary  triangula- 
tion  the  longest  possible  lines  are  to  be  obtained  which  are 
consistent  with  the  formation  of  well-proportioned  triangles. 

As  an  example  of-  one  of  the  field  computations,  suppose 
that  two  stations  28  miles  apart  are  65  and  105  feet,  respec- 
tively, above  the  ocean  level,  and  that  the  highest  point 
between  them  is  on  a  ridge  10  miles  from  the  first  station  and 
20  feet  above  ocean  level.  It  is  required  to  find  the  height 
of  towers  at  the  two  stations  so  that  the  line  of  sight  may 
pass  10  feet  above  the  top  of  the  ridge.  Assuming  that  the 
line  of  sight  is  parallel  to  a  tangent  to  the  level  surface  at 
the  ridge,  the  combined  effect  of  curvature  and  refraction 
is  0.57  X  io2  =  57  feet  for  the  first  station  and  0.57  X  i82 
=  185  feet  for  the  second  station.  Hence  the  height  of  the 
tower  at  the  first  station  should  be  57  +  30  —  65  =  22  feet, 
and  at  the  second  station  185  +  30  —  105  =  no  feet. 

Reference  may  be  made  to  the  Reports  of  the  Coast  and 
Geodetic  Survey  for  1882  and  1885  for  a  full  account  of  the 
rules  of  reconnaissance,  and  to  Final  Results  of  the  Triangu- 


200  GEODETIC   TKIANGULATION.  IX. 

lation  of  the  New  York  State  Survey  (Albany,   1887)  for  an 
interesting  description  of  the  detailed  field  work. 

After  the  reconnaissance  party  has  established  a  few  stations 
a  triangulation  party  may  start  at  work  in  the  measurement 
of  angles.  It  is  the  duty  of  this  party  to  mark  the  stations, 
erect  the  towers  and  signals,  and  make  the  observations  of 
the  horizontal  and  vertical  angles.  Sometimes  the  reconnais- 
sance and  triangulation  work  are  done  by  the  same  party, 
this  method  usually  saving  expense.  Base-line  measurement 
and  astronomical  work  are,  however,  usually  done  by  specially 
trained  parties. 

Prob.  71.  A  station  B  is  325  feet  above  A,  but  between  them,  at  a 
distance  of  15  miles  from  A  and  25  miles  from  B,  is  a  ridge  which 
is  10  feet  above  A.  If  no  tower  is  built  at  B  and  one  50  feet  high 
at  Ay  how  much  above  the  ridge  does  the  line  of  sight  pass  ? 

72.   STATIONS  AND  TOWERS. 

The  marking  of  a  station  in  a  permanent  manner  is  usually 
done  by  the  first  triangulation  party  which  takes  the  field, 
the  reconnaissance  party  merely  selecting  and  describing  the 
approximate  location.  It  is  believed,  however,  that  if  the 
responsibility  of  marking  the  station  were  assigned  to  the 
reconnaissance  party,  a  better  location  would  often  be  made. 
The  name  of  the  station  is  usually  assigned  by  the  reconnais- 
sance party,  and  this  should  be  the  same  as  the  local  name 
of  the  peak  or  ridge  on  which  it  is  situated. 

The  stations  are  marked  by  bolts  set  into  the  rock,  or  by 
stone  monuments  set  in  the  ground.  In  the  latter  case  it  is 
customary  to  bury  beneath  the  monument  a  bottle  or  crock 
whose  center  marks  the  center  of  the  station.  When  this  is 
done  the  knowledge  of  the  bottle  or  crock  should  be  con- 
cealed from  the  people  of  the  neighborhood,  and  it  should  be 
covered  with  a  large  flat  stone  having  a  hole  drilled  in  its 
upper  surface.  The  bottom  of  this  flat  stone  should  be  about 


/2.  STATIONS   AND   TOWERS.  2OI 

six  inches  above  the  crock,  its  top  about  three  feet  below  the 
surface  of  the  ground,  arid  upon  it  the  foot  of  the  monument 
may  be  set.  The,  centersr'of  the  underground  mark,  of  the 
hole  in  the  flat  stone,  and  of  the  top  of  the  monument  should 
be  in  the  same  vertical.  Near  the  top  of  the  monument 
"  U.  S.,"  or  other  appropriate  letters  should  be  cut.  Detailed 
instructions  regarding  the  methods  of  marking  stations  may 
be  found  in  the  Reports  above  cited.  Reference  points 
should  be  located  on  surrounding  rocks,  or  by  auxiliary 
monuments,  from  which  bearings  and  distances  are  to  be 
measured  to  the  station.  The  geodetic  surveyor  should 
always  make  his  description  of  the  station  clear  and  full,  so 
that  it  may  be  found  after  the  lapse  of  many  years.  For  this 
purpose  it  is  well  to  run  a  traverse  line  to  the  nearest  public 
road,  if  there  is  one  within  a  reasonable  distance,  and  erect 
there  a  monument  which  may  serve  as  a  starting  point  for 
future  parties. 

A  tower  is  a  structure  erected  over  a  station  for  the  support 
of  the  theodolite  and  observer.  It  consists  of  two  parts,  an 
interior  tripod  to  carry  the  instrument,  and  an  exterior 
scaffold  entirely  surrounding  the  tripod  but  unconnected  with 
it.  The  interior  tripod  is  usually  made  of  three  posts  braced 
together,  while  the  outside  scaffold  is  a  structure  like  a  braced 
pier  having  four  posts.  Rough  towers  made  of  timber  cut 
on  the  spot  can  be  built  for  about  $1.00  per  vertical  foot  up 
to  heights  of  30  feet,  exclusive  of  the  cost  of  the  timber. 
Towers  higher  than  50  feet  are  usually  built  of  sawn  timber 
bolted  together,  and  one  150  feet  high  makes  a  heavy  item 
in  the  expense  of  triangulation.  At  some  stations  no  towers 
are  required  as  the  instrument  may  be  directly  upon  the 
ground.  Even  in  such  cases  a  low  tower  ten  feet  in  height 
is  to  be  recommended,  as  it  adds  much  to  the  comfort  of  the 
observer  in  warm  weather,  and  has  the  advantage  of  elevating 
the  line  of  sight  above  the  surrounding  earth. 
/  The  four  posts  of  the  exterior  scaffold  should  be  extended 


202 


GEODETIC   TRIANGULATION. 


IX. 


about  eight  feet  above  the  platform  so  as  to  allow  canvas  to 
be  spread  to  protect  the  instrument  from  the  sun  and  wind. 
The  effect  of  the  sun  on  the  interior  tripod  is  in  high  towers 
often  very  marked,  the  top  moving  in  a  lateral  direction  so 
as  to  describe  an  ellipse.  To  lessen  this  effect,  and  also  for 
protection  against  wind,  it  is  often  screened  by  a  canvas 
covering  placed  around  the  upper  part  of  the  scaffold. 

The  views  of  two  triangulation  stations  of  the  U.  S.  Coast 
and    Geodetic    Survey    here    given    may    be    of    interest    to 

students.  The  first  shows  a  tower 
130  feet  in  height  erected  by 
Mosman  at  Tate,  Ohio,  and  the 
second  the  method  used  by  the 
author  at  Port  Clinton,  Pa.,  where 
no  tower  was  recuired.  In  the 


first  view  the  theodolite  is  not  visible,  but  in  the  second  it  ij 
seen  mounted  upon  the  tripod.  In  both  views  heliotropes 
for  flashing  to  the  adjacent  stations  may  be  distinguished. 
Detailed  information  regarding  the  erection  of  towers  will 
be  found  in  the  Report  of  U.  S.  Coast  and  Geodetic  Survey 
for  1882,  pages  199-208. 

Sometimes  a  church  spire,  or  other  inaccessible  point,  is 
used  as  a  station  a|nd  angles  are  measured  at  other  stations 
by  sighting  upon  it.  This  is  of  frequent  occurrence  in  second- 


72.  STATIONS  AND   TOWERS.  203 

ary  triangulation,  but  should  be  avoided  in  primary  work. 
Sometimes  in  primary  work-  an  eccentric  station  is  occupied 
near  the  true  one,  the  angles 
observed  there,  and~their  values 
then  reduced  to  the  true  sta- 
tion. Let  A  be  the  true  sta- 
tion and  a  the  eccentric  one, 
and  let  it  be  required  to  find 
the  true  angle  MAN  from  the  observed  angle  MaN.  To  do 
this  the  distance  Aa  must  be  carefully  measured  and  also  the 
angle  AaM,  and  the  distances  AM  and  AN  must  be  found 
from  the  triangulation.  Let  Aa  =  d,  AaM =  #,  AM  =  m, 
AN  =  n,  MaN  —  a,  and  MAN  =  A.  Then,  as  the  opposite 
angles  made  by  the  crossing  lines  are  equal,  A  -f-  M  equals 
a  -\-  N,  and  accordingly  the  required  angle  is 

A  =  a  -  M  +  N,  (72) 

in  which  M  and  N  are  to  be  computed  from 

s'mM  =  —  sin#,       sinN  =  -sin(0  +  a),         (72)' 
or,  for  most  primary  work,  since  m  and  n  are  large, 

M  —  206  265— sin0,       N  -  206  265  -  sin(#  +  a\       (72)" 
wi  fi 

where  M  and  N  will  be  found  directly  in  seconds.  For 
example,  let  ^—2.2145  meters,  logm  —  3  90891,  log;*  = 
3.95713,  0  =  28°  07',  a  =  64°  18'  20".  1 3.  Then  M  =  26" . 550 
and  N  —  50". 372,  whence  A  —  64°  18'  43". 95. 

The  angle  6  is  here  measured  from  the  fixed  line  aA  around 
to  the  left-hand  station  and  its  value  may  range  from  o°  to 
360° ;  hence  the  signs  of  M  and  N  will  depend  upon  the  signs 
of  sin#  and  sin(#  +  a).  Thus  if  M  and  N  were  located  at  the 
left  of  a  in  the  figure,  6  would  be  over  180°  ind  sin#  would 
be  negative.  With  regard  to  the  use  of  (72)'  and  (72)"  it 


204  GEODETIC   TRIANGULATION.  IX. 

may  be  said  that  the  former  need  not  be  employed  unless  M 
and  TV  are  greater  than  15  minutes. 

Prob.  72.  Draw  the  figure  for  the  case  where  d  '=  2.2145  meters, 
log/?/  =  2.90891,  logtf  =  2.95713,  6  =  208°  07',  a  =  96°  07'  c>3".72, 
and  compute  the  true  angle  A. 

73.  SIGNALS. 

A  signal  is  a  pole,  target,  or  other  object  erected  at  a 
station  upon  which  the  observer  at  another  station  points  in 
measuring  the  angles.  The  simplest  signal  is  a  pole,  but  its 
use  involves  a  liability  to  error  1.1  sighting  upon  the  illumi- 
nated side,  and  hence  for  the  most  accurate  work  plane 
targets  are  preferred.  These  are  made  of  a  wooden  frame- 
work covered  with  either  black  or  white  muslin.  For  a  dis- 
tance of  fifteen  or  twenty  miles  good  dimensions  for  a  target 
are  2  feet  in  width  and  12  feet  in  height.  The  target  has  the 
disadvantage  of  requiring  to  be  set  anew  whenever  the 
observer  changes  his  station,  but  it  has  the  advantage  of 
being  more  easily  seen  than  a  pole.  The  old  practice  of 
putting  a  tin  cone  on  a  pole  and  of  sighting  on  the  illuminated 
side  cannot  be  recommended,  except  for  reconnaissance  work. 
For  long  lines  neither  pole  nor  target  can  be  recognized, 
and  the  heliotrope  must  be  used.  This  instrument  consists 

essentially  of  a  mirror  which 
reflects  the  sunlight  to  the 
observer's  station.  The 
usual  size  of  the  mirror  is 
about  two  inches  in  diam- 
eter, and  it  should  be 
mounted  so  that  it  has  a 
motion  about  a  vertical  and  a  horizontal  axis.  The  mirror 
may  be  placed  at  one  end  of  a  board  about  three  feet  long 
upon  which  are  two  sights  in  the  same  line  with  the  center  of 
the  mirror.  The  sights  being  pointed  at  the  distant  station, 
the  mirror  is  constantly  turned  by  an  attendant,  called  a 


73-  SIGNALS.  205 

heliotropec,  so  that  the  shadow  of  the  rear  sight  falls  upon 
the  front  one,  and  the*  sunlight  then  is  reflected  to  the 
observer,  who  sees  jt  as  a  9*ar  twinkling  in  the  horizon.  As 
the  apparent  diameter  of  the  sun  is  about  half  a  degree,  the 
reflected  rays  form  a  cone  having  the  same  angle,  so  that  it 
is  only  necessary  to  point  the  heliotrope  within  a  quarter  of 
a  degree  of  an  object  in  order  that  the  light  may  reach  it. 
The  light  of  a  heliotrope  may  be  seen  through  haze  of 
moderate  intensity  if  the  observer  knows  where  to  point  his 
telescope  in  order  to  find  it. 

Lines  from  ten  to  fifteen  miles  in  length  are  usually 
observed  with  pole  or  target  signals.  For  lines  from  fifteen 
to  forty  miles  a  combination  of  target  and  heliotrope  is 
advantageous,  the  former  being  used  on  cloudy  days  and  the 
latter  in  sunshine;  in  this  case  the  heliotroper  erects  the 
target  over  the  station  and  places  his  instrument  in  line  in 
front  of  it.  For  lines  exceeding  fifty  miles  in  length  the 
heliotrope  is  the  only  feasible  signal  unless  the  atmosphere 
be  unusually  clear.  Probably  the  longest  side  yet  observed 
is  one  of  192  miles  in  California,  where  the  heliotrope  had  a 
mirror  of  77  square  inches. 

Night  signals  have  been  successfully  used.  These  are 
generally  large  kerosene  lamps  with  reflectors,  which  are 
placed  in  position  and  lighted  by  the  heliotropers  on  leaving 
their  stations  in  the  evening.  A  magnesium  tape  whose 
burning  is  regulated  by  clockwork  has  been  also  employed. 
Night  work  should  be  usually  combined  with  day  work,  the 
observer  being  on  duty  from  noon  to  midnight.  The  best 
time  for  measuring  horizontal  angles  is  from  six  until  nine 
o'clock  in  the  morning,  and  from  three  in  the  afternoon  until 
after  sunset,  as  then  the  air  is  the  clearest  and  the  lateral 
refraction  disturbances  are  the  smallest.  For  vertical  angles, 
on  the  other  hand,  the  best  time  is  during  the  two  hours 
preceding  and  following  noon,  the  vertical  refractiqn^ 
then  the  least  variable. 


206  GEODETIC   TRIANGULATION.  IX 

In  measuring  horizontal  angles  it  is  sometimes  necessary 
that  a  signal  should  be  set  at  a  short  distance  to  one  side  of 

the  center  of  the  station. 
This  is  called  the  case  of  an 
eccentric  signal,  and  a  correc- 
tion is  to  be  applied  to  the 
observed  angle  to  reduce  it  to 
the  true  angle.  For  instance,  in  1878  an  observer  at  the 
station  O  measured  the  angles  COa  and  aOB,  where  the 
heliotrope  had  been  set  at  a  instead  of  at  the  true  station  A. 
The  distance  Aa  was  reported  as  16  feet  2  inches,  and  the 
angle  OaA  as  129°  35'.  Later,  in  1883,  the  work  had  pro- 
gressed so  that  OA  was  found  to  be  29  556  meters.  The 
value  of  the  small  angle  AOa  in  seconds  is  computed  from 
206265^0  sinOaA/AO  and  will  be  found  to  be  26" '.63,  and 
this  is  the  correction  to  be  added  to  aOB  and  to  be  sub- 
tracted from  COa. 

An  eccentric  signal  should  be  avoided.  Indeed  it  is  best 
that  heliotropers  should  not  know  that  it  can  be  used,  other- 
wise they  will  be  often  tempted  to  set  their  heliotropes 
eccentrically  from  considerations  of  personal  comfort  and  may 
neglect  to  take  the  measurements  that  are  necessary  for  cor- 
recting the  angles. 

Prob.  73.  Compute  the  correction  AOa  when  the  side  OA  is  not 
very  large  compared  with  Aa,  say  when  OA  =  295.56  meters. 

74.  HORIZONTAL  ANGLES. 

Two  classes  of  triangulation  are  always  recognized  in 
geodetic  work;  the  primary  series,  which  connects  directly 
with  the  bases  and  has  the  longest  possible  lines,  and  the 
secondary  series,  which  locates  stations  within  the  primary 
triangles.  To  these  are  ultimately  added  a  tertiary  series  for 
establishing  stations  at  closer  intervals  for  the  special  use  of 
plane-table  and  stadia  parties.  It  is  generally  required  that 


74- 


HORIZONTAL   ANGLES. 


207 


the  probable  error  of  an  observed  value  of  a  horizontal  angle 
shall  not  exceed  o".3Q  on  primary  work  and  o/r.8o  on 
secondary  work.  On  primary  work  repeating  and  direction 
theodolites  are  used,  on  secondary  work  repeating  theodo- 
lites, while  for  the  tertiary  work  the  engineers'  transit  gives 
all  the  precision  desirable.  In  fact  a  good  engineer's  transit 
will  give  as  precise  results  as  those  required  for  secondary 
triangulation,  provided  the  length  of  the  lines.be  such  that 
the  signals  can  be  clearly  seen  with  its  telescope. 

A  repeating  theodolite  does  not  differ  in  principle  from  an 
engineers'  transit.  The  telescope,  however,  is  so  long  that 
it  cannot  be  turned  over  on  its  axis,  -but  must  be  lifted  out 
of  the  standards  in  order  to  be  reversed  in  position.  The 
graduated  limb  is  usually  from  8  to  12  inches  in  diameter,  is 
divided  into  ten-minute  divisions,  and  reads  by  three  verniers 
to  3"  or  5".  Circles  16  and  20  inches  in  diameter  were 
formerly  used,  but  it  is  now  known  that  the  precision  of 
these  is  little  if  any  superior  to  those  of  8  and  10  inches. 
The  method  of  observation,  in  order  to  eliminate  systematic 
and  accidental  errors,  is  in  all  respects  the  same  as  that 
described  in  Art.  14.  Owing  to  the 
atmospheric  disturbances  on  long  lines 
of  sight  it  is  important  that  the  work 
on  each  angle  should  be  distributed 
over  several  days,  and  this  is  easy  to 
arrange,  since  the  rarity  of  good 
weather  usually  requires  a  party  to 
remain  two  or  three  weeks  at  a  sta- 
tion when  several  lines  concentrate 
there. 

The  following  is  a  record  of  the 
work  done  with  a  repeating  theodolite 
at  Bear's  Head  station  in  Pennsylvania  from  July  20  to  July 
30,  1885.  During  these  eleven  days  there  were  only  eight 
when  the  weather  permitted  observations,  and  on  five  of 


BakeOven 


Port  Clinton 


208 


gEODETIC   TRIANGULATION. 
ANGLES  AT  BEAR'S    HEAD. 


IX. 


Name  of  Angle. 

No.    of 
Reps. 

Observed  Value. 

Adjusted. 

Penobscot  —  Knob 

48 

51"  19'  59".7i 

60''.  15 

Penobscot  —  Bake  Oven 

40 

85  52  32  .39 

33    -22 

Penobscot  —  Port  Clinton 

48 

139  24  06  .97 

06    .91 

Penobscot  —  White  Horse 

40 

1  80   39   48  .43 

47  -24 

Knob—  Bake  Oven 

48 

34   32    33  -16 

33  -07 

Knob  —  Port  Clinton 

48 

88   04   05  .75 

c6  .76 

Knob—  White  Horse 

48 

129    19   47  .57 

47  -09 

Bake  Oven  —  Port  Clinton 

48 

53    3i    34  -27 

33  -69 

Bake  Oven  —  White  Horse 

40 

94   47    12  .71 

14  .02 

Port  Clinton  —  White  Horse 

48 

41    15    39  -97 

40  -33 

these  no  measurements  could  be  made  until  about  three  o'clock 
in  the  afternoon.  The  total  number  of  measures  is  seen  to 
be  456,  .or  an  average  of  1 14  for  each  independent  angle. 
The  station  adjustment  being  made  by  the  method  of  Aft. 
16,  the  average  probable  error  of  a  single  observed  value  is 
found  to  be  o" .fi  and  that  of  an  adjusted  value  about  o".6o. 
It  is  thus  seen  that  the  adjustment  has  greatly  increased  the 
precision. 

A  direction  theodolite  has  no  verniers,  but  is  read  by  three 
or  more  micrometer  microscopes  placed  around  the  limb. 
The  circle  in  the  figure  represents  the  field  of 
view  of  one  of  the  microscopes  in  Which  three 
divisions  of  the  graduated  limb  are  seen.  By 
turning  the  micrometer  screw  the  cross-hair  is 
moved  to  a  or  b,  thus  reading  the  distance  ac  or 
be  in  seconds.  When  pointing  on  the  first  station  the  cross- 
hair may  be  set  at  a  graduation  mark,  and  when  pointing  at 
the  second  the  reading  is  taken  as  just  described.  Such 
theodolites  have  large  circles  so  that  the  limb  may  be  divided 
to  5  minutes  while  the  micrometers  will  read  to  seconds,  and 
by  taking  the  mean  of  all  the  micrometer  readings  a  close 


74-  HORIZONTAL  ANGLES.  2OQ 

determination  of  the  angle  can  be  made.  No  repetitions  are 
possible  by  this  method*  but  different  series  of  readings  are 
taken,  on  different  parts  «f  the  limb  in  order  to  eliminate 
errors  of  graduation,  measures  are  made  both  from  left  to 
right  and  from  right  to  left  in  order  to  eliminate  errors  due 
to  clamping  and  twist,  and  the  work  is  distributed  over  differ- 
ent days  to  eliminate  atmospheric  influences. 

There  are  two  methods  of  measuring  the  angles  at  a  station 
with  a  direction  theodolite.  The  first,  called  the  method  of 
single  angles,  is  to  determine  each  angle  independently  by 
the  process  above  described ;  thus  in  the  case  of  four  lines 
meeting  at  O  each  angle  is  measured  by  reading  first  on  the 
left-hand  line  and  second  on  the  right-hand  line;  thus  the 
value  found  for  BOC  or  BOD  is  independent  of  any  reading 


made  on  AOB.  In  this  method  all  the  results  are  to  be 
treated  and  adjusted  exactly  as  if  they  had  been  made  by  a 
repeating  theodolite. 

In  the  second  method  of  observation,  called  the  method  of 
directions,  a  line  OA  is  taken  as  a  reference  line  and  pointing 
and  reading  taken  on  it;  then  the  limb  is  turned  and  readings 
taken  on  B,  C,  and  D  in  succession.  Another  line  OB  is 
then  taken  as  a  reference  line,  and  readings  taken  on  C,  D, 
and  A  in  succession.  Here  it  is  seen  that  the  values  found 
are  not  independent,  as  the  initial  reading  enters  into  all  the 
results  of  each  series;  consequently  the  adjustment  is  more 
complicated  than  that  of  the  other  method. 


210  GEODETIC   TRIANGULATION.  IX. 

Prob.  74.  Regarding  the  above  observations  at  Bear's  Head  sta- 
tion as  of  equal  weight,  compute  the  probable  error  of  a  single 
observed  value. 

75.  THE  STATION  ADJUSTMENT. 

The  station  adjustment  for  all  cases  except  the  method  of 
directions  is  made  by  the  method  of  Art.  16,  which  need  not 
be  further  explained  here.  When  the  weights  are  so  nearly 
equal  as  those  of  the  case  given  in  the  last  Article,  it  is  an 
unwarrantable  refinement  to  take  them  into  account.  With 
regard  to  the  probable  errors  it  is  to  be  noted  that  those  of 
the  adjusted  values  need  rarely  be  computed  by  the  method 
of  Art.  10  except  in  special  scientific  investigations.  It  is 
well,  however,  to  find  the  probable  error  of  a  single  observa- 
tion by  formula  (10),  and  this  ought  to  have  a  reasonable 
agreement  with  the  average  probable  error  of  the  observed 
values  as  computed  from  (9)' '. 

Some  observers  prefer  to  measure  the  n  angles  included 
between  the  n  lines  meeting  at  a  station  instead  of  combining 
the  lines  to  make  \n(n  —  i)  angles  as  in  the  example  of  the 
last  Article.  This  case  is  called  "closing  the  horizon/'  and 
thus  the  conditional  equation  is  introduced  that  the  sum  of 
the  n  single  angles  shall  be  360  degrees.  The  adjustment 
may  be  made  by  the  method  of  Art.  16,  employing  only 
n  —  I  independent  quantities,  but  the  numerical  work  will 
usually  be  shorter  by  the  method  of  Art.  21. 

The  method  of  directions  requires  a  slightly  different 
process  of  station  adjustment.  To  explain  it  take  the  case 

where  the  three  lines  OA,  OB, 

\j  and    OC  meet  at  the  station   O, 

/        /$          and  let  x  and  y  be  the  most  prob- 
able  values   of    the   angles   A  OB 
and    AOC.      Suppose    that    OM 
0  denotes  the  direction  of  the  tele- 

scope when  the  mean  reading  of  the  three  microscope  microm- 


75.  THE  STATION  ADJUSTMENT.  211 

eters  is  o°  oo'  oo".oo,  and  let  m  denote  the  most  probable 
value  of  MO  A.  Then  let  the  three  readings  on  A,  B,  and  C 
give  the  three  observation  Equations 

m  =     60°  18'  20". 5, 

m  +  x  =     85    04    13   .0, 

/«  +  j  =   119    50    14  .2. 

Next  let  the  circle  be  turned  so  that  ON  gives  the  zero  direc 
tion  and  let  n  be  the  angle  NO  A.      Then  three  readings  being 
taken    on   At    B,    and    C  again,    there    are   three   additional 
observation  equations 

n  =  120°  if  05". o, 

n  +  x  -   145    02    53   .o, 

n  +  y  =   179    48    59     5. 

Again  if  the  circle  be  turned  about  60  degrees  further  and 
three  readings  be  taken  upon  A,  B,  and  C  there  will  be  three 
more  observation  equations,  while  a  fourth,  fifth,  and  sixth 
set  will  each  give  three  others.  Thus  for  six  positions  of  the 
circle  there  will  be  18  observation  equations  involving  8 
unknown  quantities;  from  these  the  normal  equations  are 
formed  by  the  rule  of  Art.  6,  or,  if  they  are  of  unequal  weight, 
by  the  rule  of  Art.  7,  and  their  solution  will  furnish  the  most 
probable  values  of  x  and  y.  The  quantities  m,  n,  .  .  .  may 
be  eliminated  from  the  normal  equations,  before  solving  for 
x  and  y,  as  their  numerical  values  are  not  required. 

The  numerical  work  may  be  abbreviated  by  introducing 
corrections  to  assumed  values  of  the  quantities.  Thus,  for 
the  above  case,  let  ml  and  nl  be  corrections  to  the  observed 
values  of  m  and  n\  also  let  ^  =  24°  45'  52". 5 -[-•*",  and 
y  =  59°  31'  5  3".  7  +.1V  Then  the  six  observation  equations 
reduce  to 

ml  —  o,          *«,  +  *,  =  o,  ml  +  y,  —  o, 

a,  =  o,  nv  +  x,  =  -  04".  5,      nl+y1  =  +  o".8, 

and  from  these  the  four  normal  equations  are 


212  GEODETIC   TRIANGULATION.  IX. 


3*.  +    *i  +    ^i  =  -  °3   -7, 
;;/,  +     ;/,  +  2*,  =   —  04   .5, 

^  +     »t  +  27,  =  +  oo   .8. 

Taking  the  values  of  ;«,  and  ;«,  from  the  first  and  second 
equations  and  substituting  them  in  the  others,  these  become 

4*i  —  2r,  =  —  09".  8,      —  2*1  +  4/i  =  +  06".  I, 

from  which  ..#,.=  —  02"  '.2$  and  j^  =  +  00^.40  are  the  most 
probable  'corrections,  whence  x  =  24°  45'  50".  2  5  and  /  = 
59°  31'  54".  10  are  the  adjusted  values  of  the  angles  AOB 
and  AOC,  and  accordingly  the  most  probable  value  of  BOC 
is  24°  36'03".85. 

The  angles  found  by  the  station  adjustment  are  spherical 
angles,  because  the  graduated  circle  is  made  level,  that  is 
parallel  to  a  tangent  plane  to  the  spheroid  at  the  station. 
Strictly  speaking  the  level  position  of  the  graduated  limb  is 
an  astronomical  and  not  a  geodetic  one  (Art.  59),  but  this 
slight  discrepancy  of  a  few  seconds  can  produce  no  measur- 
able effect  on  the  observed  angles.  It  should  be  borne  in 
mind  that  it  is  of  great  importance  to  avoid  inaccuracy  of 
level  when  measuring  angles,  since  this  renders  their  values 
too  large,  and  there  is  no  method  of  eliminating  its  influence. 

Prob.  75.  Given  the  observed  angles  AOB  =  86°  07'  17"  with 
weight  6,  BOC  —  89°  10'  35"  with  weight  4,  and  COA  — 
184°  41'  55"  with  weight'  i.  Compute  the  most  probable  values  of 
the  angles. 

76.  TRIANGLE  COMPUTATIONS. 

After  the  angles  have  been  measured  at  a  number  of 
stations  and  the  length.  of  one  side  has  been  obtained,  either 
by  connecting  with  an  adjacent  triangulation  or  by  measuring 
it  as  a  base,  computations  of  the  lengths  of  the  triangle  sides 
are  to  be  made.  The  three  angles  of  a  triangle  do  not  add 
up  to  1  80  degrees  and  hence  the  results  obtained  for  the  sides 


76.  TRIANGLE   COMPUTATIONS.      .  213 

are  only  approximate,  but  they  are  more  than  sufficiently 
accurate  to  compute  the  spherical  excess  of  the  triangle. 
These  computations  are  tile  same  in  every  respect  as  those 
explained  in  Art.  19,  except  that  five-place  logarithms  should 
be  used,  the  logarithmic  sines  taken  to  the  nearest  10"  of 
angle,  and  the  lertgths  determined  only  to  the  nearest  10 
meters./ 

The  formula  for  spherical  excess  established  in  (6$)'  of  Art. 
63  may  now  be  used  and  the  excess  be  found  for  each  tri- 
angle. In  order  to  take  the  factor  m  from  Table  IV  the 
mean  latitude  must  be  known  roughly.  In  the  first  instance 
this  may  be  estimated,  but  in  later  work  it  will  be  found  from 
the  results  of  the  LMZ  computations.  Then, 

Spherical  excess  —  m .  ab  sindT, 

in  which  a  and  b  are  any  two  sides  of  the  triangle  and  C  is 
the  angle  included  between  them. 

As  a  numerical  example  of  the  computation  of  spherical 
excess  the  following  data  of  a  triangle  will  be  used : 


Stations. 

Angles  adjusted 
at  Stations. 

Approximate 
Distances. 

Approximate 
Latitudes. 

Pimple  Hill 

49° 

04' 

50' 

'.13 

27 

540 

meters 

4i° 

02' 

Smith's  Gap 

9° 

21 

25 

•53 

36 

440 

meters 

40 

49 

Bake  Oven 

40 

33 

46 

.91 

23 

700 

meters 

40 

45 

Sum  =  180°  po'  02''. 57  Mean  L.  =  40°  52' 

Now  C  can  be  taken  as  any  one  of  these  angles  and  a  and  b 
as  the  two  adjacent  sides.  It  is  advisable  to  make  two  check 
computations  for  the  excess,  thus: 

Numbers.           Logarithms.  Numbers.           Logarithms. 

factor  m             9.40441  factor  m             9.40441 

a  =  36  440             4-56158  <z  =  23  700             4-37475 

b  =  23  700             4-37475  b  —  27  540            4.43996 

C  =  49°  04'  50"  1.87831  C  =  9D°  21'  30"  1.99999 

Excess=  oi".66            0.21905  Excess  =  01".  66            0.21911 

The  adjustment  of  the  angles  of  a  spherical  triangle  is  to 


214  GEODETIC    TRIANGULATION.  IX- 

be  made,  when  the  angles  are  of  equal  weight,  by  applying 
to  each  of  the  given  angles  one-third  of  the  discrepancy 
between  the  theoretic  sum  and  the  actual  sum  (Art.  18). 
For  instance,  using  the  above  triangle,  the  correction  to  be 

Stations                     Angles  adjusted  Spherical  Plane 

at  Stations.  Angles.  Angles. 

Pimple  Hill          49°  04'  5o".i3  49".%3  49".28 

Smith's  Gap         90    21    25    .53  25    .22  24    .66 

Bake  Oven            40    33    46    .91  46    .61  46   .06 

Sum  =  180°  oo'  02".57  oi".66  oo".oo 

1 80°  -f  Excess  =  1 80    oo   01    .66 


Discrepancy  =  —  oo    .91 

subtracted  from  each  given  angle  is  oo".3O,  and  thus  are 
found  the  adjusted  spherical  angles  whose  sum  is  180° 
OOf  oi".66.  Then,  to  find  the  plane  angles  between  the 
chords  of  the  spherical  arcs,  one-third  of  oi/r.66  is  subtracted 
from  each  spherical  angle. 

When  the  weights  of  the  given  angles  are  very  unequal  it 
is  advisable  to  take  them  into  account  by  the  method  of  (18). 
Thus  if  the  triangle  KPS  have  8  sets  measured  at  K  and  48 
at  both  P  and  5,  and  if  the  computed  spherical  excess  is 
oi".83,  the  spherical  angles  are  found  by  applying  corrections 


Stations.  Weights.  Angc 


K  i  41°  20'  34"-34  35"-52  34"-9I 

P  6  79    03    41    .73  41    .93  41    .32 

S  6  59    35    44    .18  44    -38  43    -77 

Sum  =  180°  oo'  oo".25  oi".83  oo".oo 
180°  +  Excess  =  180    oo    01    .83 
Discrepancy  =             +O1    -58 

inversely  proportional  to  the  weights,  and  then  the  plane 
angles  are  derived  by  diminishing  each  spherical  angle  by  one- 
third  of  the  excess. 

After  these  adjustments  have    been    completed    a    second 
computation  of  triangle  sides  is  to  be  made  with  seven-place 


76. 


TRIANGLE   COMPUTATIONS. 


215 


logarithms  and  using,  of  course,  the  plane  angles  just  found. 
The  method  is  in  all  respects  identical  with  that  exemplified 
in  Art.  19.  The  following,«form,  which  may  be  used  for  this 
computation,  shows  the  angles  at  the  stations,  the  adjusted 
spherical  angles,  and  the  plane  angles,  the  spherical  excess 
for  this  case  being  02" .82.  The  triangle  sides  thus  computed 
are  the  lengths  of  the  spherical  arcs  on  the  surface  of  the 
spheroid,  the  length  of  the  base  having  been  reduced  to  that 
surface  by  the  method  of  Art.  31. 

COMPUTATION   OF  A   SPHERICAL  TRIANGLE. 


Lines  and 
Stations. 

Angles  at 
Stations. 

Corr. 

Spherical 
Angles. 

Sph. 
Excess. 

Distances  and 
Plane  Angles. 

Logarithms. 

AB 

43075    -54 

4.6342308 

C 

54°  58'  08".  84 

-f-o".o8 

08".  92 

—  o"-94 

07".  98 

0.0868007 

A 

95   29  01  .87 

+  0  .08 

01  .96 

—  o  .94 

01    .02 

1.9980079 

B 

29  32  51  .86 

+  0    .08 

51    -94 

—  o  .94 

51  .00 

1.6929746 

CB 

52364    .80 

4.7190394 

CA 

25942    .16 

4.4140061 

• 

The  spherical  angles  here  determined  give  the  azimuths  of 
AC  and  BC  when  the  azimuths  for  the  other  side  are  known. 
Thus,  if  the  azimuths  of  AB  and  BA  are  204°  10'  36". 05  and 
24°  14'  07". 92,  the  azimuth  of  AC  is  299°  39'  38". 01  and 
that  of  BC  is  350°  41'  15". 98.  If  the  latitude  and  longitude 
of  A  and  B  are  known,  the  two  LMZ  com- 
putations for  finding  the  latitude  and  longi- 
tude of  C  and  the  azimuths  of  CA  and  CB 
may  now  be  made  by  the  method  of  Art. 
66,  the  logarithms  of  the  lengths  of  the 
sides  being  transferred  from  the  above 
form.  Then  the  next  triangle,  having  AC 
or  BC  as  its  base,  may  be  treated  in  like 
manner,  and  thus  from  one  measured  base 
and  one  astronomical  station  a  chain  of 
adjusted  and  computed. 


2l6  GEODETIC   TRIANGULATION.  IX. 

Prob.  76.  Make  all  the  computations  described  in  this  Article  for 
the  data  of  the  following  triangle,  taking  the  angles  as  of  equal 

Stations  Angles  adjusted  Approximate 

at  Stations.  Latitudes. 

Knob  50°  37'    17". 20  40°   54' 

Bake  Oven  98     37     05    .05  40     45 

Smith's  Gap  30     45     41     .35          .       40     40 

weight,  the  length  of  the  side  opposite  Knob   as  27535.63   meters, 

and  the  azimuth  from  Bake  Oven  to  Smith's  Gap  as  252°  26'  55". 42. 

YT.  THE  FIGURE  ADJUSTMENT. 

There  are  two  classes  of  conditions  to  be  satisfied  in  the 
adjustment  of  a  geodetic  triangulation,  those  arising  at  the 
stations  and  those  arising  from-  the  geometry  of  the  figure. 

The  station  adjustment  has 
already  been  discussed,  and 
now  the  figure  adjustment  is  to 
receive  attention.  This  figure 
adjustment  gives  rise  to  condi- 
tions of  two  kinds,  called  angle 
conditions  and  side  conditions. 
The  requirement  that  the  sum  of  the  adjusted  spherical 
angles  of  a  triangle  shall  equal  180  degrees  plus  the  spherical 
excess  is  an  angle  condition,  while  the  requirement  that  the 
length  of  any  side  shall  have  the  same  value  by  whatever 
route  it  be  computed  is  a  side  condition.  For  instance,  in 
the  figure  ABCS  there  are  four  station  adjustments  to  be 
made,  if  the  angles  are  measured  at  the  four  stations;  then, 
the  figure  adjustment  requires  that  three  angle  conditions  and 
one  side  condition  shall  be  satisfied. 

The  strict  method  of  making  the  adjustment  of  the  case 
shown  in  the  above  figure  is  to  state  observation  equations 
involving  the  angles  at  the  stations,  and  conditional  equations 
involving  all  the  requirements  of  both  station  and  figure 
adjustments.  Thus,  if  three  angles  be  measured  at  each 
station  there  will  be  twelve  observation  equations;  for  illus- 


77-  THE   FIGURE   ADJUSTMENT.  217 

tration  suppose  the  angles  have  been  measured  only  to  the 
nearest  degree,  and  that  the  observation  equations  are 

,4  =  84°,  ^,=40°,  4  =  43°>*  B=  56°,  B,=  30°,  *.=  27°, 
C=40°,  £>i9°>  Ct  =  2i°,  5,  =  106°,  £,=  135°,  S.=  i20°, 

where  5,  is  the  angle  subtended  by  AB,  and  5,  and  St  those 
subtended  by  BC  arid  CA.  Then,  the  station  adjustments 
give  the  four  conditions 

A=A,+A, ,  B=Bl+B* ,  C=C,+C, ,  S, +£,+£,  =  360°. 
The  figure  adjustment,  supposing  the  triangles  to  be  plane 
ones,  requires  the  three  angle  conditions 

^,+^+5,=  1 80°,      ^  +  C,+S,=  180°,      ^+^1+5,=  i8o°, 
and  also,  as  shown  in  Art.  22,  the  side  condition 
sin//!  sin^,  sin£7,  =  sinA^  sin.Z?2  sin(72. 

The  problem  now  is  to  determine  the  most  probable  values 
of  the  twelve  observed  angles  which  at  the  same  time  satisfy 
the  eight  conditional  equations. 

This  problem  is  capable  of  rigorous  solution,  but  when  a 
figure  contains  many  triangles  it  leads  to  very  laborious  com- 
putations. The  custom  has  hence  arisen  of  dividing  the  work 
into  two  parts;  first,  the  station  adjustments  are  made,  each 
independently  of  the  others,  and  secondly  the  values  found 
by  these  station  adjustments  are  then  corrected  so  as  to 
satisfy  all  the  conditions  of  the  figure  adjustment.  The 
station  adjustments  are  generally  made  in  the  field,  but  the 
figure  adjustment,  which  is  far  more  lengthy,  is  reserved  for 
the  office,  and  is  made  by  the  method  of  correlates  that  is 
explained  in  the  next  Article. 

In  applying  these  principles  to  a  triangle  net  consisting  of 
a  chain  of  simple  triangles,  having 
one  side  AB  measured  as  a  base  and         B 
all  angles  observed,  it  is  seen  that 
the  figure  adjustment  has  as  many 
angle  equations  as  there  are  triangles,  and  no  side  equations. 


2l8  GEODETIC   TRIANGULATION.  IX. 

The  figure  adjustment  is  hence  very  simple,  each  triangle 
being  treated  in  succession  by  the  method  of  Art.  76,  and 
the  spherical  angles  and  plane  angles  thus  found  are  the  final 
adjusted  values.  If,  however,  another  side  HK  be  also 
measured  as  a  base,  then  a  conditional  side  equation  is  to  be 
introduced  to  express  the  requirement  that  the  length  HK  as 
computed  from  AB  shall  be  the  same  as  the  measured  length; 
an  illustration  of  this  case  for  two  triangles  is  given  in  the 
last  paragraph  of  Art.  22. 

In  primary  geodetic  triangulation  all  stations  are  occupied 
and  all  lines  sighted  over  in  both  directions.  In  secondary 
work  a  few  of  the  stations  may  not  be  occupied,  these  being 
church  spires  or  other  inaccessible  points.  Thus  in  the  last 
figure  if  one  of  the  stations  between  B  and  H  be  not  occupied 
the  number  of  angle  equations  will  be  diminished  by  three, 
because  there  will  be  three  triangles,  in  each  of  which  one 
angle  has  not  been  observed  and  hence  its  value  is  to  be 
found  from  those  of  the  observed  angles. 

In  stating  the  conditional  equations  that  enter  into  a  figure 
adjustment  care  should  be  taken  to  introduce  no  unnecessary 
ones,  and  the  following  rules  will  be  useful  for  that  purpose; 
these  rules  suppose  only  one  base  to  have  been  measured. 
Let  n  be  the  total  number  of  lines  and  nf  the  number  of  lines 
sighted  over  in  both  directions,  let  s  be  the  total  number  of 
stations  and  s'  the  number  of  stations  occupied  for  angle 
measurements.  Then,  in  the  figure  adjustment, 

Number  of  angle  equations  =  »'  —  /  -j-  i,         .     , 
Number  of  side  equations      =  n  —  2s  +  3. 

For  instance,  in  the  figure  ABCS,  at  the  beginning  of  this 
Article,  n'  =  n  =  6,  s'  =  s  =  4,  and  hence  there  are  three 
angle  equations  and  one  side  equation;  if,  however,  the 
station  S  had  not  been  occupied,  then  n'  =  3,  n  =  6,  A'  =  3, 
s  =  4,  and  accordingly  there  would  be  one  angle  equation 
and  one  side  equation. 


78.  CONDITIONED   OBSERVATIONS.  2 19 

P/ob.  77.  How  many  angle  and  side  equations  are  there  in  the 
figure  adjustment  of  each  of,  the  triangle  nets  shown  in  Art.  24,  one 
base  and  the  angles  being  measured  7 

78.  CONDITIONED  OBSERVATIONS. 

By  the  proper  selection  of  the  unknown  quantities  it  is 
generally  possible  to  state  observation  equations  so  that  these 
quantities  will  be  independent  (Art.  25),  but  a  shorter  method 
of  adjustment,  known  as  the  method  of  correlates,  may  be 
established.  In  this  method  each  observed  quantity  is  repre- 
sented by  a  letter  and  all  the  conditional  equations  are 
written,  as  in  the  illustration  of  the  last  Article.  Let  x,  y, 
z,  etc.,  represent  the  quantities  whose  values  are  to  be  found, 
and  let  the  conditional  equations  be 

*.*  +  &*y  +  -  -  •  =  #> 
bs  +  b.y  +  .  .  .  =  b, 

c&  +  c*  y  +  •  -  •  =  c* 

in  which  the  coefficients  and  constant  terms  are  theoretic 
numbers.  Now  let  M}  ,  M^ ,  Mt ,  ...  be  the  values  found 
by  the  observations  for  x>  y,  zy  .  .  . ;  if  these  values  be  in- 
serted in  the  conditional  equations  they  will  not  reduce  to 
zero,  owing  to  the  errors  of  the  measurements.  Hence,  let 
v\ ,  ^a ,  ^3 ,  .  .  .  be  small  corrections  which  when  applied  to 
Ml ,  J/a ,  M9 ,  .  .  .  will  render  them  the  most  probable  values. 
Then  if  x,  y,  .  .  .  be  replaced  by  Ml-\-vlt  Mt  +  vt ,  .  .  .  the 
conditional  equations  reduce  to 


in  which  dl ,  dt ,  dt ,  .  .  .  are  small  quantities  called  discrepan- 
cies. The  problem  now  is  to  find  values  of  vt ,  z/, ,  vt ,  .  .  . 
which  exactly  satisfy  these  equations  and  which  at  the  same 
time  are  the  most  probable  values. 


220  GEODETIC   TRIANGULATION.  IX. 

The  following  is  the  solution  of  this  problem  which  is 
deduced  in  treatises  on  the  Method  of  Least  Squares.  Let 
Pi »  A »  A  »  •  •  •  De  tne  weights  of  the  observations  M{  ,  J/s , 
J/, ,  .  .  .  and  let  £,  ,  £a ,  kt ,  .  .  .  be  quantities  which  are 
determined  by  the  solution  of  the  normal  equations 


rtf 3~]  I  ab~^  rac  ~\ 

l-jl*.  +  Lj]*.  +  l-j]*.  +  '-'  =  4. 

LAP  +  ' ' '  =  d"      (78)' 


P**IA     i  -  i  A     '  —  i  A 

-r  *•  -I- 1  r  J*.  H- 1  T  I* 


/ 


These  equations  are  the  same  in  number  as  the  number  of 
conditional  equations,  £,  being  known  as  the  correlate  of  the 
first  equation,  £a  of  the  second,  and  so  on.  The  brackets 
indicate  summation  in  accordance  with  the  same  notation  as 
that  employed  in  Art.  7,  namely, 

^i — ^L__j-f^         \ab~\  - a^  i  *A  i 

/J-A"hA  "  L/JSS  A  "  A   + 

and  the  coefficients  have  similar  properties  to  those  in  the 
normal  equations  for  independent  observations. 

By  the   solution  of  these  normal  equations   the  values  of 
kl  ,  k^ ,  £, ,  .  .  .  are  found;   then  the  corrections  are 


and  these  added  to  Ml  ,  J/a ,  .  .  .  give  the  most  probable 
values  of  AT,  jj/,  ...  which  exactly  satisfy  the  theoretic  con- 
ditions. When  there  is  but  one  conditional  equation  there 
is  but  one  normal  equation  and  one  correlate,  k^  whose  value 

is  d.j  I  —    ,  and  thus  the  values  of  z/, ,   z/3 ,  .  .  .  agree  with 


78.  CONDITIONED   OBSERVATIONS.  221 

those  deduced  in  Art.  21,  where  •dl  is  called  d,  and  q  is  used 
instead  of  a.  f   * 

As  an  illustration  of  the  n&ethod,  let  there  be  five  measure- 
ments on  five  quantities,  giving  the  observation  equations, 

1.  x  =  47.26,      with  weight     3, 

2.  y  —  39-O4,     with  weight  19, 

3.  z  =     6.35,      with  weight  13, 

4.  w=  86.64,     with  weight  17, 

5.  u  =  35.21,      with  weight    6, 
which  are  subject  to  the  two  theoretical  conditions, 

x  +  y  —  w  =  o,         y  -\-  z  —  u  =  10. 

Let  vl  ,  v^ ,  z/s ,  v^ ,  and  z/§  be  the  most  probable  corrections  - 
to   the  observed   values,   so  that  the    observation  equations 
become 

i  '  Jr\    ~          J , 

2.  z>a  =  O,  p^  =    19, 

3-  v,  =  o,                 /,  =  13, 

4-  ^  =  o,                 /4  =  17, 

5-  ^  =  o,                /.  =  6, 
and  the  conditional  equations  reduce  to 

Vl  +  Vt  -  v,'  =  +  0.34, 

v*  +  vs  -  v,  =  -  0.18. 

Now,  by  comparison  with  the  notation  in  (78), 

and  thus  the  normal  equations  of  (78)'  become 

0.445^  +  0.053^  =  +  0.34, 

°-°53^i  +  0.296^  =  —  o.  18, 

whose  solution  gives  k,  =  -(-0.855  and  k^——  0.759.      Then 
by  (78)"  the  values  of  the  corrections,  or   residual  errors,  are 

and  hence  the  adjusted  values  of  the  observations  are 
•*  =  47-545,    7  =  39-045,     3  =  6.291,     w  =  86.590,     u  =  35.336, 


222  GEODETIC   TRTANGULATION.  IX. 

which  are  the  most  probable   results  that  exactly  satisfy  the 
two  conditional  equations. 

The  probable  error  of  an  observation  of  the  weight   unity 
may  be  computed  by  the  formula 


rl  =  0.6745*7       ^      ,. 

Y   n  -  q  +  n> 


in  which  n  is  the  number  of  observation  equations,  q  the 
number  of  unknown  quantities,  and  n'  the  number  of  condi- 
tional equations.  For  the  above  example  the  residuals  are 
already  found;  squaring  them,  multiplying  each  by  its 
weight,  and  adding,  gives  ^pv1  =  0.428,  whence  r^  =  0.309. 
Accordingly  the  probable  error  of  the  first  observation  is 
0.309/4/3  =  0.18,  and  the  weight  of  the  adjusted  value  of 
that  observation  must  be  somewhat  smaller  than  o.  18. 

Prob.  78.  Four  lines  OA,  OBy  OC,  and  OD  meet  at  a  station  O, 
and  the  following  angles  are  observed,  all  of  equal  weight  : 
AOB  =  19°  47'  13",  BOC  =  40°  38'  04",  COD  =  65°i2'io", 
DO  A  =  54°  22'  2<j',BOD  =  105°  50'  i6",JDOA  =  119°  30' 42". 
Let  x,  y,  z,  and  w  represent  the  four  angles  first  named.  Compute 
their  adjusted  values  by  the  method  of  correlates. 

« 

79.  ADJUSTMENT  OF  A  POLYGON. 

Let  the  diagram  represent  a  polygonal  figure  having  an 
interior  station  S,  and  let  the  angles  which  each  side  makes 

with  the  line  to  5  be  measured, 
5  being  an  unoccupied  station. 
By  applying  the  rule  of  the  last 
Article  it  is  seen  that  there  are 
two  conditions  in  the  figure 
adjustment,  one  angle  equation 
*B  and  one  side  equation.  If  the 

figure  be  a  plane  one  the  angle  condition  is  that  the  sum  of 
the  ten  interior  angles  when  adjusted  shall  be  540  degrees 
for  a  five-sided  polygon.  The  side  equation  results  from  the 


79-  ADJUSTMENT   OF  A   POLYGON.  223 

condition  that,  if  one  side  be  computed  from  another  by  two 
routes,  the  two  expressions  for  its  value  shall  be  equal. 

To  express  the  first  condition  algebraically  let  ^, ,  vt, 
.  .  .  z>10  be  corrections  in  seconds  to  be  added  to  the  observed 
values  of  the  angles.  Let  dl  be  the  difference  in  seconds 
between  the  theoretic  sum  and  the  sum  of  the  ten  observed 
values;  then 

^  +  *',  +  ^3  +  "4  +  V,  +  V<  +  V,  +  V%  +  V,  +  V19  =  d, 

is  the  conditional  angle  equation.  To  state  the  second  con- 
dition let  expressions  for  the  side  SD,  as  computed  from  SA 
•by  two  routes,  be  written;  if  these  be  equated  there  results 

sin^,  sin£s  sinC5  sinZ>,  sin^  =  sin^2  sin#4  sinC6  sinZ>8  sin£10 
as  the  conditional  side  equation.  This  is  to  be  expressed  in 
terms  of  the  corrections  in  a  similar  manner  to  that  used  in 
Art.  22,  log  (A,  +  v,)  being  written  as  log  A,  +  vt  diff.  i" ', 
where  diff.  \"  is  the  tabular  logarithmic  difference  of  the 
logarithmic  sine  corresponding  to  the  angle  Alt 

Let  the  observed  values  be  those  written  below,  all  being 
of  equal  weight.  As  the  sum  of  these  is  540°  oo'  10"  the 
discrepancy  dl  is  —  10",  the  angle  equation  is  known,  while 

Observed  Angles.  Log.  Sines. 

Al  =  25°   47'    23"  1  6385588  +  43.52;, 

£3  —  56     31     22  1.9212208  -f  I4.ov3 

C*.  =  85     28     57  1.9986487+     i.7v§ 

A  =  83     12     39  1.9969439+     2.5z/7 

-^9  =  4i      l6     15  1.8192933  +  23.9^ 

1.3746655 

At  =  50°  12'  54"  1.8856162  +  17.52;, 

BI  —  48  52   12  1.8769214  +  18.4^ 

C%  =  61  58  02  1.9458027  -f-  n.2vt 

£>*  =  38  25  °7  1-7933543  +  26.52;. 

^io=48  15   19  1.8728079  -f  i8.8z;10 

l-3745°25 

the  side  equation  is  found  by  equating  the  two  sums  of  the 
logarithmic  sines.  Thus 


224  GEODETIC   TRIANGULATION.  IX. 

»1      4-     *B       4"      *l      4-       7'4     +     Vf,     4-       7'8      +    Z/7     +       »8      4-      7/j       4      «/,„       =  — lo", 

43-5t'i-I7'5z'94-i4-°*'s— 18.4^4+1  .jr,,— 11.2^,4-2.5^  — 26.57/84-23.9^9  — i8.8z/IO=- 1630, 

where  the  second  member  of  the  last  equation  is  in  units  of 
the  seventh  decimal  place  of  logarithms. 

By  the  method  of  the  last  Article  the  solution  is  now 
readily  made,  placing  #,  =  -f-  I,  #a  =  -(-  i,  .  .  .  d^  =  —  10 
and  b,  =  +  43.5,  ^  =  —  17.5,  .  .  .  ^a  =  +  1630.  The  two 
correlative  normal  equations  are  found  to  be 

-|-  io/£,  —  6.8/£a  =:  —  10,  —  6.8/£,  -j-  4494.5^  =  --  1630, 
from  which  £,  =  —  1.248  and  £a  =  —  0.364.  Then  by  (78)" 

^=-17".!,     *.  =  -6".3,     »§=-i".9,     *,=-*"•*,     *.=-9".9, 

»,  =  +  5"-i,     *4=+5"-5,     ^  =  +  2".8,     ^.  =+8".4,     ^10-  +  5"  6, 

are  the  most  probable  corrections  to  the  observed  values,  and 
applying  them  the  sum  of  the  adjusted  values  will  be  found 
to  be  exactly  540°,  and  then  the  angles  at  5  may  be  obtained. 
Also,  multiplying  each  v  by  its  tabular  difference,  the  correc- 
tions to  the  logarithms  may  be  found,  and  the  sums  of  the 
two  sets  should  then  be  exactly  equal. 

For  a  large  polygon  where  the  spherical  excess  of  the  tri- 
angles can  be  detected  the  method  of  adjustment  is  the 
same,  the  two  conditional  equations  being  slightly  modified. 
First,  the  theoretic  sum  of  the  ten  angles  exceeds  540  de- 
grees by  two-thirds  of  the  spherical  excess  of  the  entire  poly- 
gon; thus  if  this  excess  be  i8".o  the  discrepancy  dl  will  be 
i8".o  —  io".o  =•+  8".o.  Secondly,  each  observed  angle  is 
to  be  diminished  by  one-third  of  the  spherical  excess  of  its 
triangle  before  placing  it  in  the  side  equation ;  for  instance, 
if  the  spherical  excess  of  the  triangle  ABS  is  03". o,  then  the 
value  of  Al  to  be  used  in  the  side  equation  is  25°  47'  22" . 
The  solution  is  now  made  as  before  and  the  corrections 
vl ,  v^ ,  .  .  .  z>10  found;  these,  added  to  the  angles  used  in 
the  side  equations,  give  the  adjusted  plane  angles,  or,  added 
to  the  observed  values,  they  give  the  adjusted  spherical 
angles. 


80.  ADJUSTMENT   OF   A   QUADRILATERAL.  225 

When  the  station  5  is  occupied  and  all  the  angles  there  are 
observed  there  will  be  .five  angle  equations  and  one  side 
equation  in  the  figure  adjustment.  The  side  equation  is  the 
same  as  before  and  the  five  angle  equations  may  be  taken  as 
those  expressing  the  conditions  that  the  sum  of  the  angles  in 
each  triangle  shall  equal  its  theoretic  value.  Thus,  for  the 
triangle  ABS,  if  the  sum  of  the  observed  angles  be  180°  oo' 
05"  and  the  spherical  excess 'be  03". o,  the  angle  equation  is 
v*  +  ^s  +  vn  =  +02".  These  six  equations  lead  to  six 
correlative  normal  equations,  by  whose  solution  the  six  cor- 
relatives are  found,  and  then  the  fifteen  corrections  are 
obtained.  Lastly,  the  adjusted  spherical  angles  result  by 
adding  these  corrections  to  the  observed  values,  and  the 
adjusted  plane  angles  are  found  by  subtracting  from  the 
spherical  angles  the  proper  amount  for  spherical  excess.  It 
may  be  remarked,  however,  that  this  solution  can  be  abbre- 
viated by  an  artifice  similar  to  that  used  in  the  next  Article. 

Prob.  79.  In  the  first  diagram  of  Art.  77  let  there  be  given 
Al  =  4oi°,  At  =  43i°,  B,  =  29f°,  B%  =  26|°,  C,  =  19°,  C,  =  21°. 
Adjust  these  observations  so  that  the  results  shall  satisfy  all  the  con- 
ditions of  the  figure  adjustment. 

80.  ADJUSTMENT  OF  A  QUADRILATERAL. 

In  the  quadrilateral  ABCD  let  the  two  single  angles  at 
each  corner  be  equally  well  measured.  The  rule  of  Art.  77 

shows  that  the  figure    adjustment  c 

requires  three  angle  equations  and 
one  side  equation.  The  three  angle 
equations  may  be  written  by  taking 
any  three  of  the  triangles  and  im- 
posing the  conditions  that  in  each 
the  sum  of  the  adjusted  values  shall  equal  the  theoretic  sum; 
the  three  triangles  that  have  the  point  B  in  common  will  be 
chosen  for  this  purpose.  Let  dl ,  d^ ,  d^  be  the  discrepancies 
for  these  triangles,  d^  being  that  for  the  triangle  whose  large 


226  GEODETIC   TRIANGULATION.  IX. 

angle  is  A,  while  d,  and  d^  are  those  for  the  triangles  whose 
large  angles  are  B  and  C\  also  let  d^  be  the  discrepancy  for 
the  fourth  triangle  CD  A.  Let  vl  ,  v,  ,  .  .  .  v%  be  the  correc- 
tions in  seconds  to  be  applied  to  the  observed  values.  Then 
the  three  conditional  angle  equations  are 


*>•     +     *>.     +    %     +     «>.     =     ^3    , 

and  the  conditional  side  equation  is 

sin^4t  sin^B  sin(77  sinZ\  =  sin^4,  sin^4  sin£"6  sinZ>8. 
For  given  numerical  values  of  the  eight  angles  the  adjustment 
may   now   be  made   by  the  method  of  Art.  78,  there  being 
four  correlatives  and  four  normal  equations. 

It  is,  however,  frequently  required  to  make  an  approximate 
adjustment,  whereby  the  three  angle  equations  will  be  satis- 
fied, the  side  equation  not  being  used.  Taking,  then,  three 
correlatives,  the  normal  equations  (78)'  are,  since  all  weights 
are  unity, 

4k,  +  2k,  =  d,  , 

2k,   +  4^  +  2k,   =  d,  , 

2k,  +  4k,  =  d,  . 

Solving  tliese  and  substituting  the  values  in  (78)",  the  cor- 
rections are  found,  and,  remembering  that  d,  -\-  -d<  =  */,  +  dt  , 
these  may  be  written 

*>>  =  v,  -  K  +  M  -  4), 


which  are  very  easy  in  numerical  application.  For  instance, 
let  the  three  triangles  have  the  spherical  excesses  .$•,  =  O7/.48, 
st  =  i".05,  ss  —  ix/.4i,  and  st  =  0^.84,  and  let  the  observed 
values  of  the  eight  angles  be  arranged  in  four  sets,  one  for 
each  triangle.  The  sum  of  the  observed  angles  for  the  first 
set  subtracted  from  the  theoretic  sum  gives  the  discrepancy 


80.  ADJUSTMENT  OF  A   QUADRILATERAL.  22? 

dl  =  -J-O2". 73,  and  similarly  for  the  other  sets.  Then  by 
(80)  the  corrections  are  ;-{- o".62,  -f-°"'74»  -j-  o".67,  and 
-|- o".56,  whence  result  the  ^adjusted  values  of  the  spherical 
angles.  The  sum  of  these  is  360°  oo'  oi".88,  which  is  a 
check  on  the  work,  since  the  spherical  excess  of  the  figure  is 
jj  -|-  ja  =  j,  -|-  J4  =  I'/.SQ,  the  error  of  one  unit  in  the  second 
decimal  being  due  to  the  lost  digits  in  the  third  decimal. 
Lastly,  the  plane  angles  are  found  by  subtracting  the  proper 
amounts  from  the  spherical  angles,  these  amounts  being  com- 
puted from  (80)  by  using  the  given  excesses  instead  of  the 
discrepancies. 


Observed 

Di  =    58  '  44' 
At  =    25     18 
Aa  =     58     54 
B*  =     37    02 
Bs  =    27    38 

a  =  56  24 

a  =     33    53 

Angles. 
A 
38".98 
16  .80 
57   .54 
04  -43 

B 

57"-54 
04  .43 
46  .48 
09  .77 

C 

09    .77 

35   -14 

D 
16  .80 

35  -14 

Adjusted  Angles. 
Spherical.        Plane. 
39".  60           39.51 
17  .42         17.33 
58    .28           58.14 
05    .17           O5.O2 
47  .15         46.77 
10  .44         10.06 
35  -70         35.38 

Dt,  =    62 

03 

27 

•  56 

27 

•  56 

28    .12            27.80 

179 

59 

57  -75 

58 

.22 

58 

•95 

58 

.48 

01  .88        00.01 

180 

00 

oo  .48 

or 

•05 

01 

.41 

oo 

.84 

01  .89        oo.oo 

-{-  O2    .73     -f-  02    .83     -\-  O2    .46     -j-  O2    .36  OO    .OO  OO.OO 

The  adjusted  values  thus  found  will  not,  in  general,  satisfy 
the  side  equation,  but  by  the  following  process  a  second  series 
of  corrections  may  be  obtained  that  will  insure  this  result. 
Let  z/j ,  z/a ,  .  .  .  z/§  be  the  additional  corrections  to  be  applied 
to  the  above  values  of  the  plane  angles.  Then  the  angle 
equations  are 

Vi  +  Vi  +  Vi  +  V*   =   °> 

v,  +  v,  +  v,  +  v.  =  o, 

^i  +  ^.  +  ^  +  ^8  =  o, 
and  the  side  equation  takes  the  form 

aj,\  +  aj>^  +  a,v,  +  atv,  —  a,v,  —  a.vt  —  a,v,  —  atve  =  </, 
where  al ,  #, ,  .  .  .  a%  are  the  tabular  differences  of  the  loga- 


228  GEODETIC   TRIANGULATION.  IX. 

rithmic  sines  corresponding  to  the  values  of  the  plane  angles 
and  d  is  the  difference  between  the  sums  of  the  logarithmic 
sines  of  the  even  and  odd  angles.  Now  let 


v%  =         u,  —  u,  ,          ut  =        u,  —  u4, 
v,  =  —  u,  +  u9,          v,  =  -  u,  +  ut  , 

Vt    =    —    «,    —    W,  ,  7;7    =     —    «,    —    «B  , 

and  thus  the  angle  equations  are  satisfied,  while 

(<*,  H-  ««  +  «.  +  «,  —  «,  —  «,--  «B  —  08K  +  (*i  H-  *e)«. 

+  (*»  —  «i)«s  +  K  +  ^K  +  (—«,  +  «,)«B  =  ^ 

is  the  side  equation  in  terms  of  the  new  quantities.  This 
may  be  treated  by  the  method  of  Art.  78,  and  after  the  «'s 
are  found,  the  values  of  v  are  known.  For  the  above 
numerical  case  this  side  equation  becomes 


57-4^,  -  15-2^3       54-2«4  —  20.  i«§  =  +  3, 

where  the  second  member  is  in  units  of  the  seventh  decimal 
place  of  the  logarithms.  Then  the  single  correlative  equa- 
tion gives  k,  =  0.000288,  whence  #,  =  -|-o".ooi7,  and  the 
other  u's  are  smaller  still;  accordingly  the  corrections  i\  ,  v9t 
.  .  .  v9  do  not  in  any  case  amount  to  one  one-hundredth  of  a 
second.  The  final  adjusted  values  of  the  angles  are  hence 
those  above  given;  had  the  corrections  due  to  the  side  equa- 
tions been  appreciable  they  would  have  been  added  to  both 
spherical  and  plane  angles  in  order  to  give  the  final  adjusted 
results. 

Prob.  80.  All  the  angles  at  stations  A,  By  and  C  are  measured,  but 
none  at  D.  Find  the  number  of  equations  in  the  figure  adjustment 
and  state  them. 

81.  FINAL  CONSIDERATIONS. 

The  preceding  principles  will  enable  the  student  to  adjust 
and  compute  any  common  triangle  net  having  but  one  meas- 
ured base.  If  the  net  be  composed  of  triangles  only,  each 


8l.  FINAL   CONSIDERATIONS.  229 

succeeding  the  other,  the  adjustment  is  made  by  starting  with 
the  base  and  computing,  each  triangle  in  succession  by  the 
method  of  Art.  76.  If  it  b&  composed  of  polygons  only,  each 
is  separately  adjusted  by  Art.  79,  and  the  triangle  sides  then 
computed  from  the  plane  angles.  If  it  be  composed  of 


quadrilaterals  only,  the  adjustment  of  each  is  made  by  Art. 
80,  and  the  sides  then  found  from  the  plane  angles.  If  it  be 
made  up  of  triangles,  polygons,  and  quadrilaterals,  as  is 
generally  the  case,  the  same  process  may  be  followed,  by 
starting  at  the  base  and  treating  each  part  in  succession. 
Lastly  the  latitudes,  longitudes,  and  azimuths  are  computed 
by  the  method  of  Art.  66  if  the  triangulation  be  a  geodetic 
one,  or  by  that  of  Art.  19  if  it  be  plane. 

When  two  bases  are  measured,  or  two  stations  occupied  for 
astronomical  work,  the  adjustment  becomes  so  complex  that 
it  cannot  be  discussed  in  this  elementary  book.  Such  adjust- 
ments can  only  be  successfully  done  by  an  office  force 
specially  trained  in  precise  computation,  and  several  weeks 
are  perhaps  required  to  solve  the  correlate  normal  equations 
that  arise;  this  work  is  to  be  carried  out  in  a  systematic 
manner  so  that  constant  checks  on  the  accuracy  of  the 
numerical  work  may  be  secured,  and  the  probable  errors  of 
the  final  adjusted  values  may  be  determined. 


230  GEODETIC   TRIANGULATION.  IX. 

It  is  not  difficult  to  measure  a  single  angle  so  that  the  final 
mean  shall  have  a  probable  error  as  low  as  o".5,  but  the 
probable  error  of  the  adjusted  value  of  the  same  angle  as 
found  from  the  figure  adjustment  will  usually  be  somewhat 
larger.  In  this  case  the  comparison  of  the  probable  errors 
does  not  perhaps  give  the  fully  correct  idea,  for  there  can  be 
no  doubt  but  that  the  figure  adjustment  has  been  most 
useful  in  eliminating  accidental  errors  due  to  the  pointings  on 
signals,  which  in  the  measurement  of  a  single  angle  may 
perhaps  be  a  constant  source  of  error.  Undoubtedly  the 
precision  of  the  results  of  the  figure  adjustment  is  materially 
greater  than  that  of  the  angles  as  determined  by  the  station 
adjustment. 

Throughout  the  entire  field  and  office  work  all  the  coordi- 
nates, distances,  and  azimuths  are  to  be  regarded  as  approxi- 
mate until  the  final  figure  adjustment  is  finished  and  the  LMZ 
computations  based  on  these  are  completed.  In  the  begin- 
ning of  the  field  work  the  latitudes  and  longitudes  are  known 
very  roughly,  being  taken  from  such  maps  as  are  available, 
or  found  from  compass  readings  and  estimated  distances. 
Later  they  become  known  to  within  a  minute  from  rough 
triangle  and  LMZ  computations,  and  at  the  close  of  the 
season's  field  work  the  various  determinations  of  a  station 
should  check  within  a  tenth  of  a  second.  After  the  office 
work  of  adjustment  is  finished,  however,  the  latitudes  and 
longitudes  will  agree  to  thousandths  of  a  second,  and  then 
the  triangle  net  may  be  regarded  as  definitely  completed. 

During  the  progress  of  the  field  work  vertical  angles  are 
often  taken  by  the  method  briefly  described  in  Chapter  IV. 
Such  angles  are  to  be  measured  by  an  instrument  having  a 
full  vertical  circle  so  that  the  double  altitude,  or  double  zenith 
distance,  may  be  obtained  by  reversal.  This  vertical  angle 
work,  although  its  results  cannot  compare  in  precision  with 
that  of  spirit  leveling,  furnishes  valuable  information  in  a  new 
country  which  will  repay  its  slight  cost. 


8l.  FINAL   CONSIDERATIONS.  2$  I 

Boundary  lines  between  countries  are  run  most  accurately 
after  a  triangulation  hast  covered  a  strip  along  the  general 
route.  The  location  of  the,  stations  being  thus  determined, 
that  of  the  boundary  line  is  computed  from  the  principles  of 
geodesy  and  then  points  are  set  upon  it  by  running  out 
traverses  from  the  ,  stations.  Many  boundary  lines  have 
been  run  by  determining  astronomically  the  latitudes  and 
longitudes  of  stations  and  then  running  out  traverses  and 
deflection  lines  from  these.  Owing,  however,  to  the  uncer- 
tainty of  the  plumb-line  deflections  these  boundaries  cannot 
compare  in  precision  with  those  determined  from  a  geodetic 
triangulation. 

Prob.  81.  Consult  Report  of  the  Commission  on  the  Survey  of  the 
Northern  Boundary  of  the  United  States  (Washington,  1878,)  and 
explain  how  points  were  located  on  the  49th  parallel  of  north 
latitude. 


THE   FIGURE   OF   THE   EARTH. 


CHAPTER  X. 
THE    FIGURE    OF   THE    EARTH. 

82.  THE  EARTH  AS  A  SPHEROID. 

In  the  preceding  chapters  the  fundamental  principles  for 
determining  the  size  and  eccentricity  of  the  spheroid  that 
best  represents  the  earth  have  been  presented,  and  the 
methods  for  computing  geodetic  surveys  for  given  spheroidal 
elements  have  been  explained.  Now  in  conclusion  a  few 
other  methods  will  be  briefly  discussed  by  which  the  dimen- 
sions or  oblateness  of  the  spheroid  may  be  determined. 

Pendulum  observations  give  information  regarding  the 
ellipticity  of  the  spheroid,  since  the  length  of  a  pendulum 
beating  seconds  is  proportional  to  the  force  of  gravity  and 
since  this  force  is  greater  in  the  polar  than  in  the  equatorial 
regions.  Clairaut  in  1743  deduced  a  remarkable  theorem  for 
the  length  of  the  seconds'  pendulum  at  any  latitude,  namely, 

s  =  S  +  ($k  -  f)S  sin'Z,  (82) 

in  which  s  is  the  length  at  the  latitude  L,  and  5  is  the  length 
at  the  equator,  k  the  ratio  of  the  centrifugal  force  at  the 
equator  to  the  force  of  gravity,  and  /  the  ellipticity  of  the 
earth  regarded  as  an  oblate  spheroid.  This  theorem  is  limited 
only  by  the  assumptions  that  the  earth  is  a  spheroid  rotating 
on  its  axis,  and  that  its  material  is  homogeneous  in  each  of 
the  concentric  spheroidal  strata.  Now  if  the  values  of  5  and 
(^k  —  f}S  can  be  found  from  observations,  then  \k  —  /is 
known,  and  since  from  the  principles  of  mechanics  k  can  be 
closely  ascertained,  the  ellipticity /is  determined. 


82.  THE   EARTH   AS   A   SPHEROID.  233 

For  example,  the  following  are  a  few  of  the  many  observa- 
tions that  have  been  made  on  the  length  of  the  seconds' 
pendulum:  ^ 

Place.  Latitude.  Length  of  Pendulum. 

Spitzbergen  79°  49'  58"  39.2147  inches 

Hammerfest  70  40  05  39.1952 

London  51  31  08  39 .1 393 

New  York  40  42  43  39-TOI7 

Jamaica  17  56  07  39-°35T 

Sierre  Leone  8  29  28  39.0200 

St.  Thomas  o  24  41  39.0207 

For  each  of  these  observations  there  may  be  written  an 
observation  equation  of  the  above  form;  letting  T  represent 
the  coefficient  of  sina/,  the  first  one  is 

39.2147  =  S  +  0.968847; 

and  similarly  for  each  of  the  others.  Then,  applying  the 
Method  of  Least  Squares,  the  most  probable  values  of  5  and 
T  are  found  to  be  39.0155  and  0.2021  inches.  Accordingly 
the  ratio  of  T  to  S  is  0.005181,  and  this  is  the  value  of 
|-/£  — f.  But  the  value  of  k  is  about  -g-J-y  as  found  from  the 
known  facts  regarding  the  intensity  of  gravity  and  the 
velocity  of  rotation  at  the  equator;  consequently  the  value 
of/  is  about  ^J^-.  Numerous  discussions  of  pendulum  obser- 
vations appear  to  lead  to  the  conclusion  that  the  ellipticity  of 
the  earth,  considered  as  a  spheroid,  is  not  far  from  ^--g^.T  or 
^^.  This  is  slightly  larger  than  the  value  found  from  the 
discussions  of  meridian  arcs,  and  the  conclusion  must  hence 
be  drawn  that  probably  the  spheroidal  strata  are  not  strictly 
homogeneous. 

A  theoretic  discussion  by  Newton  of  the  form  assumed  by 
a  rotating  homogeneous  fluid  under  the  action  of  gravity  and 
centrifugal  force  led  to  the  conclusion  that  the  ellipticity  was 
y^-Q.  A  similar  one  by  Laplace  indicates  that  f  is  about  ^T. 
This  value  is,  however,  far  too  great,  and  it  is  accordingly 
indicated  that  the  earth  was  not  an  homogeneous  fluid  at  the 


234  THE   FIGURE   OF  THE   EARTH.  X. 

time  it  assumed  the  present  shape.  For  a  full  exposition  of 
this  branch  of  the  subject  reference  is  made  to  Todhunter's 
History  of  the  Theories  of  Attraction  and  of  the  Figure  of 
the  Earth,  London,  1873. 

The  shape  of  the  earth  may  also  be  found  from  astronomi- 
cal observations  and  computations.  Irregularities  in  the 
motion  of  the  moon  were  first  explained  by  the  deviation  of 
the  earth  from  a  spherical  form,  and  then  these  irregularities 
being  precisely  measured,  the  ellipticity  may  be  computed, 
the  value  found  by  Airy  being  ^T,  which  is  a  little  smaller 
than  the  result  deduced  from  meridian  arcs. 

The  size  of  the  spheroid  may  also  be  deduced  from 
measured  arcs  of  a  parallel  between  points  whose  longitudes 
are  known.  It  is  evident  that  such  arcs  have  a  special  value 
in  determining  whether  or  not  the  equator  and  the  parallels 
are  really  circles.  The  field  work  of  a  triangulation  net 
extending  across  the  American  continent  along  the  parallel 
of  39°  north  latitude  was  nearly  completed  in  1899  and  the 
results  of  its  discussion  will  soon  be  available.  It  may  be 
noted,  finally,  that  the  elements  of  the  spheroid  may  be 
deduced  from  a  single  geodesic  line  whose  end  latitudes  and 
azimuths  have  been  observed,  or  from  such  a  line  derived 
from  a  geodetic  triangulation.  The  discussion  of  a  geodesic 
line,  extended  through  the  Atlantic  states  from  Maine  to 
Georgia,  by  the  U.  S.  Coast  and  Geodetic  Survey,  indicates 
that  its  influence  upon  our  knowledge  of  the  figure  of  the 
earth  is  to  increase  but  slightly  the  dimensions  of  Clarke's 
spheroid  of  1866  without  appreciably  changing  his  value  of 
the  ellipticity. 

Three  hundred  and  fifty  years  ago,  when  men  first  began 
to  think  about  the  shape  of  the  earth  on  which  it  was  their 
privilege  to  live,  they  called  it  a  sphere,  and  they  made  rude 
measurements  on  its  great  surface  to  ascertain  its  size.  These 
measurements,  after  nearly  two  centuries  of  work,  reached  an 


83.  THE   EARTH   AS   AN   ELLIPSOID.  235 

extent  and  precision  sufficient  to  prove  that  its  surface  was 
not  spherical.  Then  the  earth  was  assumed  to  be  a  spheroid 
of  revolution,  and  with  the  lapse  of  time  the  discrepancies  in 
the  data,  when  compared  on  that  hypothesis,  proved  also  that 
the  assumption  was  incorrect.  Granting  that  the  earth  is  a 
sphere,  there  has  been  found  the  radius  of  one  representing  jt 
more  closely  than  any  other  sphere;  granting  that  it  is  a 
spheroid,  there  has  been  also  found,  from  the  best  existing 
data  combined  in  the  best  manner,  the  dimensions  of  one  that 
represent  it  more  closely  than  any  other  spheroid.  It  has 
been  seen  that  the  radius  of  the  mean  sphere  could  only  be 
found  by  first  knowing  the  elliptical  dimensions,  and  here  it 
may  be  also  thought  that  the  best  determination  of  the  most 
probable  spheroid  would  be  facilitated  by  some  knowledge  of 
the  theory  of  the  size  and  shape  of  the  earth  considered  under 
forms  and  laws  more  complex  than  those  thus  far  discussed. 
In  the  following  Articles,  then,  there  will  be  given  some 
account  of  the  present  state  of  scientific  knowledge  and 
opinion  concerning  the  earth  as  an  ellipsoid  with  three 
unequal  axes,  the  earth  as  an  ovaloid,  and  lastly  the  earth 
as  a  geoid. 

83.  THE  EARTH  AS  AN  ELLIPSOID. 

As  the  sphere  is  a  particular  case  of  the  spheroid,  so  the 
spheroid  is  a  particular  case  of  the  ellipsoid.  The  sphere  is 
determined  by  one  dimension,  its  radius;  the  spheroid  by 
two,  its  polar  and  equatorial  diameters;  while  in  the  ellipsoid 
there  are  three  unequal  principal  axes  at  right  angles  to  each 
other  that  establish  its  form  and  size.  As  in  the  spheroid,  the 
ellipsoid  meridians  are  all  ellipses,  but  the  equator  instead  of 
being  a  circle  is  an  ellipse  of  slight  eccentricity.  Let  #,  and 
#a  denote  the  greatest  and  least  semi-diameters  of  the  equator 
of  the  ellipsoid,  and  b  the  semi-polar  diameter;  the  ellipti- 
cities  of  the  greatest  and  least  meridian  ellipses  then  are 

a,  —  b  #,  —  b 

=  -—      and          =  -— 


236 


THE    FIGURE   OF   THE    EARTH. 


X, 


while  all  other  meridian  ellipses  have  intermediate  values. 
For  the  equator  the  ellipticity  is  (a,  —  a^/a,.  When  the 
values  of  al ,  a^ ,  and  b  are  known,  the  dimensions  and  pro- 
portions of  the  ellipsoid  and  of  all  its  sections  are  fully 
determined. 

Qn  an  ellipsoidal  earth  the  curves  of  latitude,  with  the 
exception  of  the  equator,  are  not  plane  curves,  and  hence 

cannot  properly  be  called 
parallels.  This  results  from 
the  definition  of  latitude  as 
may  be  seen  from  the  dia- 
gram, where  PP  is  the  polar 
axis,  PQPQ  the  greatest 
meridian  section,  A  a  place 
of  observation  whose  horizon  is  AA  and  latitude  ABQ.  Let 
now  the  least  meridian  ellipse,  projected  in  the  line  PP,  be 
conceived  to  revolve  around  PP  until  it  coincides  with  the 
plane  PQPQ  and  becomes  seen  as  PQ'PQ '.  To  find  upon  it 
a  point  A'  that  shall  have  the  same  latitude  as  A,  it  is  only 
necessary  to  draw  a  tangent  A' H'  parallel  to  AH  touching 
the  ellipse  at  A',  then  A' B'  perpendicular  to  A' H'  makes  the 
same  angle  with  the  plane  of  the  equator  QQ  as  does  AB. 
If  the  lea^t  meridian  section  be  now  revolved  back  to  its  true 
position,  A'  becomes  projected  at  D' .  Therefore,  while  a 
section  through  A  parallel  to  the  equator  is  an  ellipse  ADA, 
the  curve  joining  the  points  having  the  same  latitude  as  A  is 
not  plane,  but  a  line  of  double  curvature  AD' A. 

The  process  for  determining  from  meridian  arcs  an  ellipsoid 
to  represent  the  figure  of  the  earth  does  not  differ  in  its 
fundamental  idea  from  that  explained  in  the  last  chapter  for 
the  spheroid.  The  normal  to  the  ellipsoid  at  any  point  will 
usually  differ  slightly  from  the  actual  vertical  as  indicated  by 
the  plumb  line,  and  the  sum  of  the  squares  of  these  devia- 
tions is  to  be  made  a  minimum  in  order  to  find  the  most  prob- 
able elements  of  the  ellipsoid.  An  expression  for  the  differ- 


83.  THE    EARTH   AS   AN   ELLIPSOID.     ^  237 

ence  of  these  deviations  at  two  stations  on  the  same  meridian 
arc  is  first  deduced  in  terms  of  four  unknown  quantities,  three 
being  the  semi-axes  al ,  a^  ^and  b,  or  suitable  functions  of 
them,  and  the  fourth  the  longitude  of  the  greatest  meridian 
ellipse,  referred  to  a  standard  meridian  such  as  that  of  Green- 
wich; and  in  terms  of  four  known  quantities,  the  observed 
linear  distance  between  the  two  stations,  their  latitudes  and 
the  longitude  of  the  arc  itself.  Selecting  now  one  station  in 
each  meridian  arc  as  a  point  of  reference,  there  are  written 
for  that  arc  as  many  equations  as  there  are  latitude  stations, 
inserting  the  numerical  values  of  the  observed  quantities. 
These  equations  will  contain  four  more  unknown  letters  than 
there  are  meridian  arcs,  and  from  them  as  many  normal 
equations  are  to  be  deduced  as  there  are  unknown  quantities, 
and  the  solution  of  these  will  furnish  the  most  probable  values 
of  the  semi-axes  al ,  a^ ,  and  $,  with  the  longitude  of  the 
extremity  of  al  ,  and  also  the  probable  plumb-line  deviations 
at  the  standard  reference  stations.  The  process  is  long  and 
tedious,  but  it  is  easy  to  arrange  a  system  and  schedule,  so 
that  the  computations  may  be  accurately  carried  out  and 
constant  checks  be  furnished. 

The  first  deduction  of  an  ellipsoid  to  represent  the  figure 
of  the  earth  was  made  in  Russia,  by  Schubert,  about  the  year 
1859.  He  found/j  =  ^fa  and /a  =  ^-J^  for  the  two  meridian 
ellipticities,  and  -g^y  for  that  of  the  equator.  The  longitude 
of  the  ellipse  of  greatest  eccentricity  was  found  to  be  about 
41°  East  of  Greenwich,  and  the  length  of  its  quadrant  was 
determined  as  10002263  meters,  that  of  the  quadrant  of 
least  eccentricity  being  10001  707  meters. 

It  is,  however,  Clarke  of  the  British  Ordnance  Survey  to 
whom  is  due  the  credit  of  the  most  careful  investigations  in 
this  direction.  His  discussion  of  1866  included  meridian  arcs 
amounting  in  total  to  nearly  five-sixths  of  a  quadrant  and 
containing  40  latitude  stations,  while  his  discussion  of  1878 
included  the  same  data  and  several  additional  arcs,  The  fol- 


238 


THE   FIGURE   OF   THE   EARTH. 


X. 


lowing  table  gives  the  result  of  these  discussions.  The  linear 
dimensions  are  in  meters,  the  meter  used  being  one  equiva- 
lent to  3.28086933  feet  and  hence  slightly  longer  than  the 
U.  S.  legal  meter  that  has  been  employed  in  the  preceding 
Chapters. 

CLARKE'S    ELEMENTS  OF  THE   ELLIPSOID. 


1866. 

1878. 

Greatest  equatorial  semi-diameter,  a\ 

6  378  294 

6378  380 

Least  equatorial  semi-diameter,  aa 

6376350 

6377916 

Polar  semi-axis,  b 

6356068 

6356397 

Greatest  meridian  quadrant,  q\ 

10  ooi  553 

10  ooi  867 

Least  meridian  quadrant,  q* 

10  000024 

10001  507 

Quadrant  of  the  equator,  Q 

10017475 

10018  770 

Greatest  meridian  ellipticity,  f\ 

1/287.0 

1/289.5 

Least  meridian  ellipticity,  f 

I/3I4-4 

1/295.8 

Ellipticity  of  the  equator,  F 

1/3281 

1/13706 

Longitude  of  gt 

15°  34'  East 

8°  15'  West 

In  comparing  these  ellipsoids  the  different  positions  of  the 
greatest  meridian  attract  notice;  in  the  first  it  passes  through 
Germany  and  in  the  second  through  Ireland,  the  angular  dis- 
tance between  them  being  about  24  degrees.  The  least 
meridian,  90  degrees  distant,  passes  through  Pennsylvania 
for  the  first  spheroid  and  through  Kansas  for  the  second.  It 
would  thus  appear  that  the  radius  of  curvature  is  greater  on 
the  American  than  on  the  European  continent. 

It  seems  to  be  the  prevailing  opinion  that  satisfactory 
elements  of  an  ellipsoid  to  represent  the  earth  cannot  be 
obtained  until  geodetic  surveys  shall  have  furnished  more  and 
better  data  than  are  now  available,  and  particularly  data  from 
arcs  of  longitude.  The  ellipticities  of  the  meridians  differ  so 
slightly  that  measurements  in  their  direction  alone  are  insuffi- 
cient to  determine,  with  much  precision,  the  form  of  the 
equator  and  parallels.  In  Europe  and  America  several  longi- 


84.  THE   EARTH   AS   AN   OVALOID.  239 

tude  arcs  will  soon  be  available,  and  it  will  then  be  possible 
to  obtain  more  reliable  elements  of  the  spheroid.  At  present 
the  ellipsoids  represent  the  figure  of  the  earth  as  a  whole  very 
little  better  than  do  the  spheroids,  although,  for  certain  small 
portions,  they  may  have  a  closer  accordance.  For  instance, 
the  average  piobable  error  of  a  plumb-line  deviation  from  the 
normals  to  the  Clarke  ellipsoid  of  1866  is  i".35,  while  for  the 
spheroid  derived  from  the  same  data  it  is  i".42.  Further, 
the  marked  differences  in  the  ellipticities  of  the  equator  of  the 
two  Clarke  ellipsoids,  due  to  comparatively  slight  differences 
in  data,  are  not  pleasant  to  observe.  Lastly,  the  ellipsoid  is 
a  more  inconvenient  figure  to  use  in  calculations  than  the 
spheroid.  For  these  reasons  the  earth  has  not  yet  been 
regarded  as  an  ellipsoid  in  practical  geodetic  computations, 
and  it  is  not  probable  that  it  will  be  for  a  long  time  to  come. 

84.  THE  EARTH  AS  AN  OVALOID. 

In  a  spherical,  spheroidal,  or  ellipsoidal  earth  the  northern 
and  southern  hemispheres  are  symmetrical  and  equal;  that  is 
to  say,  a  plane  parallel  to  the  equator,  at  any  south  latitude, 
cuts  from  the  earth  a  figure  exactly  equal  and  similar  to  that 
made  by  such  a  plane  at  the  same  north  latitude.  The 
reasons  for  assuming  this  symmetry  seem  to  have  been  three: 
first,  a  conviction  that  a  homogeneous  fluid  globe,  and  hence 
perhaps  the  surface  of  the  waters  of  the  earth,  must  assume 
such  a  form  under  the  action  of  centrifugal  and  centripetal 
forces;  secondly,  ignorance  and  doubt  of  any  causes  that 
would  tend  to  make  the  hemispheres  unequal;  and  thirdly, 
an  inclination  to  adopt  the  simplest  figure,  so  that  the  labor 
of  investigation  and  calculation  might  be  rendered  as  easy  as 
possible.  These  reasons  are  all  very  good  ones,  but  gradually 
there  have  arisen  certain  considerations  leading  to  the  conclu- 
sion that  there  are  causes  which  tend  to  make  the  southern 
hemisphere  greater  than  the  northern.  These  considerations 


240  THE  FIGURE   OF  THE   EARTH.  X. 

embrace  a  vast  field  of  inquiry  in  astronomy  and  physical 
geography  of  which  only  a  brief  statement  can  be  given  here. 
The  earth  moves  each  year  in  an  ellipse,  the  sun  being  in 
one  of  the  foci,  and  revolves  each  day  about  an  axis  inclined 
some  66^  degrees  to  the  plane  of  that  orbit.  When  this  axis 
is  perpendicular  to  a  line  drawn  from  the  center  of  the  sun  to 
that  of  the  earth  occur  the  vernal  and  autumnal  equinoxes, 
and  at  points  equally  removed  from  these  are  the  summer  and 
winter  solstices.  For  many  centuries  the  earth's  orbit  has 
been  so  situated  in  the  ecliptic  plane  that  the  perihelion,  or 
nearest  point  to  the  sun,  has  nearly  coincided  with  the  winter 
solstice  of  the  northern  hemisphere  and  the  summer  solstice 
of  the  southern  hemisphere.  The  consequences  are:  first, 
the  half  of  the  year  corresponding  to  the  winter  is  about  seven 
days  longer  in  the  southern  hemisphere  than  in  the  northern; 
secondly,  during  the  year  the  south  pole  has  about  170  more 
hours  of  night  than  of  day,  while  the  north  has  about  I/O 
more  hours  of  day  than  of  night;  and,  thirdly,  winter  in  the 
northern  hemisphere  occurs  when  the  sun  is  at  his  least  dis- 
tance from  the  earth,  and  in  the  southern  when  he  is  at  his 
greatest.  From  these  three  reasons  it  would  seem  that  the 
amounts  of  heat  at  present  annually  received  by  the  two 
hemispheres  should  be  unequal,  the  northern  having  the  most 
and  the  southern  the  least.  Now,  on  considering  the  physi- 
cal geography  of  the  globe,  these  two  facts  are  seen:  first, 
fully  three-fourths  of  the  land  is  in  the  northern  hemisphere 
clustered  about  the  north  pole,  while  the  waters  are  collected 
in  the  southern;  and  secondly,  the  south  pole  is  enveloped 
and  surrounded  by  ice  to  a  far  greater  extent  than  the 
northern.  There  is  then  a  considerable  degree  of  probability 
that  some  connection  exists  between  these  astronomical  and 
terrestrial  phenomena,  that  the  former,  indeed,  may  be  the 
cause  of  the  latter.  The  mean  annual  temperature  of  the 
southern  hemisphere  may  have  been  for  many  centuries 
sufficiently  lower  than  that  of  the  northern  hemisphere  to 


84.  THE   EARTH   AS  AN   OVALOID.  24! 

have  caused  an  accumulation  of  ice  and  snow  whose  attraction 
drags  the  waters  toward  it,  thus  leaving  dry  the  northern 
lands  and  drowning;  the  southern  ones  with  great  oceans. 
Hence  there  appear  to  be  causes  which  tend  to  render  the 
earth  ovaloidal,  or  egg-like,  in  shape,  the  large  end  being  at 
the  south  pole. 

The  process  of  finding  the  dimensions  of  an  ovaloid  of 
revolution  to  represent  the  figure  of  the  earth  would  be  the 
same  in  principle  as  that  already  described  for  the  spheroid 
and  ellipsoid.  The  equation  of  an  oval  should  be  stated,  and 
preferably  one  that  reduces  to  an  ellipse  by  the  vanishing  of 
a  certain  constant.  From  this  equation  an  expression  for  the 
length  of  an  arc  of  the  meridian  for  both  north  and  south 
latitude  can  be  deduced,  and  this  be  finally  expressed  in  terms 
of  the  small  deviations  between  the  plumb  lines  and  the 
normals  to  the  ovaloidal  meridian  section  at  the  latitude 
stations.  The  solution  of  these  equations  by  the  Method  of 
Least  Squares  will  give  the  most  probable  values  of  the  con- 
stants, determining  the  size  and  shape  of  the  oval  due  to  the 
data  employed,  Such  computations  have  not  yet  been 
undertaken,  on  account  of  the  lack  of  sufficient  data  from 
geodetic  surveys  in  the  southern  hemisphere.  Since  such 
surveys  can  only  be  executed  on  the  continents  and  largest 
islands,  it  is  clear  that  the  data  will  always  be  few  in  number 
compared  with  those  from  the  northern  hemisphere.  Pendu- 
lum observations,  discussed  on  the  hypothesis  of  a  spheroidal 
globe,  by  Clairaut's  theorem,  are  able  to  give  some  informa- 
tion; since  they  can  be  made  on  small  islands  as  well  as  on 
the  main  lands,  it  is  possible  thereby  to  obtain  knowledge 
concerning  the  separate  ellipticities  of  the  two  hemispheres. 

An  important  idea  to  be  nofed  in  this  branch  of  our  subject 
is  that  the  surface  of  the  waters  of  the  earth  is,  probably, 
not  fixed,  but  variable.  About  the  year  1250,  the  perihelion 
and  the  northern  winter  solstice  coincided,  and  the  excess  in 
annual  heat  imparted  to  the  northern  hemisphere  was  near  its 


242  THE   FIGURE   OF   THE    EARTH.  X. 

maximum.  Since  that  date  they  have  been  slowly  separat- 
ing, and  are  now  nearly  eleven  degiees  apart.  This  separa- 
tion increases  annually  by  about  6i".75,  so  that  a  motion  of 
180  degrees  will  require  about  10450  years,  and  when  that  is 
accomplished  the  perihelion  will  coincide  with  the  southern 
winter  solstice.  Then  the  condition  of  things  will  be  exactly 
reversed,  and  perhaps  the  ice  will  accumulate  around  the 
north  pole,  the  waters  will  flow  back  from  the  south  to  the 
north,  and  the  lands  in  the  southern  hemisphere  become  dry 
while  those  in  the  northern  hemisphere  become  submerged. 
The  change  is  so  slow  that  it  might  remain  undetected  for 
centuries  and  yet  ultimately  be  sufficient  to  perceptibly  alter 
the  relative  shapes  and  sizes  of  the  two  hemispheres.  These 
considerations,  though  interesting,  are  speculations  only,  for 
causes  not  now  known  may  intervene  to  produce  results  which 
as  yet  have  not  been  even  faintly  imagined. 

85.  THE  EARTH  AS  A  GEOID. 

The  word  Geoid  is  used  tD  designate  the  actual  figure  of 
the  surface  of  the  waters  of  the  earth.  The  sphere,  the 
spheroid,  the  ellipsoid,  the  ovaloid,  and  many  other  geometri- 
cal figures  may  be,  to  a  less  or  greater  degree,  sufficient  prac- 
tical approximations  to  the  geoidal  or  earthlike  shape,  yet  no 
such  assumed  form  can  be  found  to  represent  it  with  perfect 
accuracy.  The  geoid,  then,  is  an  irregular  figure  peculiar  to 
our  planet;  so  irregular,  indeed,  that  some  have  irreverently 
likened  it  unto  a  potato;  and  yet  a  figure  whose  form  may 
be  said  to  be  subject  to  fixed  physical  laws,  if  only  the  funda- 
mental ideas  implied  in  the  name  be  first  clearly  and  mathe- 
matically defined. 

The  first  definition  is  that  the  surface  of  the  geoid  at  any 
point  is  perpendicular  to  the  direction  of  the  force  of  gravity, 
as  indicated  by  the  plumb  line  at  that  point;  from  the  laws 
of  hydrostatics  it  is  evident  that  the  free  surface  of  all  waters 


85. 


THE    EARTH   AS   A   GEOID. 


243 


in  equilibrium  must  be  parallel  to  that  of  the  geoid.  The 
second  definition  determines  that  our  geoidal  surface  to  be 
investigated  is  that  coincidiog  with  the  surface  of  the  great 
oceans,  leaving  out  of  consideration  the  effects  of  ebb  and 
flood,  currents  and  climate,  wind  and  weather.  Under  the 
continents  and  islands  this  surface  may  be  conceived  to  be 
produced  so  that  it  shall  be  at  every  place  perpendicular  to 
the  plumb-line  directions.  If  a  tunnel  be  driven  on  the  sur- 
face from  ocean  to  ocean  it  is  evident  that  the  water  flowing 
from  each  would  attain  equilibrium  therein  and  its  level 
would  show  the  form  of  the  geoid  along  that  section  of 
the  earth. 

The  following  figure  may  perhaps  give  a  clearer  idea  of  the 
properties  of  the  geoid  and  of  its  relation  to  the  spheroid.  It 
represents  a  small  part  of  a  meridian  section,  the  northern  part 

rP 


being  land  and  the  southern  part  being  the  ocean.  The  full- 
line  curve  shows  the  section  of  the  spheroid,  while  the  lighter 
line  shows  that  of  the  geoid.  At  any  station  »S  the  line  SN 
is  normal  to  the  spheroid,  while  the  line  SZ  is  the  direction 
of  the  plumb  line  or  of  the  force  of  gravity.  Hence  the  sur- 
face of  the  g-eoid  fs  norrrml  nt  each  station  to  the  line  SZ. 
The  line  PP  being  drawn  parallel  to  the  axis  of  the  earth  and 
QQ  parallel  to  the  equator,  it  is  clear  that  the  angle  which 
SZ  makes  with  QQ  is  the  astronomical  latitude  of  the  station 
S,  while  the  angle  which  SN  makes  with  QQ  is  the  geodetic 
latitude.  The  angle  ZSN  is  hence  the  difference  of  these 
latitudes,  or  the  so-called  plumb-line  deflection. 


244  THE   FIGURE   OF   THE   EARTH.  X, 

The  figure  represents  roughly  the  probable  relative  positions 
of  the  spheroid  and  geoid.  Under  the  continents  the  geoid 
tends  to  rise  higher,  while  on  the  oceans  it  tends  to  sink  lower, 
than  the  surface  of  a  spheroid  of  equal  volume.  The  attrac- 
tion of  the  heavier  and  higher  continents  lifts,  so  to  speak, 
the  geoidal  surface  upward,  while  the  lower  and  lighter  ocean 
basins  allow  it  to  sink  downward.  To  this  rule  there  are, 
however,  many  exceptions,  and  these  exceptions  teach  that 
the  earth's  crust  is  of  variable  density;  for  instance,  south- 
ward of  the  great  mountains  in  India  it  would  be  expected 
that  the  plumb-line  deflections  would  all  be  toward  the  north, 
but  this  is  by  no  means  the  case. 

It  may  now  be  seen  that  the  plumb-line  deflections  are 
really  something  artificial,  depending  upon  the  use  of  a  par- 
ticular spheroid.  The  geoid  is  an  actual  existing  thing;  the 
spheroid  is  not,  but  is  largely  an  assumption  introduced  for 
practical  and  approximate  purposes.  At  any  station  vS  in  the 
above  figure,  the  direction  SZ  is  the  only  one  that  can  be 
observed,  and  the  angle  made  by  it  with  QQ  can  be  measured 
with  a  probable  error  of  less  than  one-tenth  of  a  second  of  arc. 
The  angle  ZSN,  or  the  so-called  plumb-line  deflection  at  5, 
will  hence  vary  with  the  elements  of  the  particular  spheroid 
employed,  and  with  the  correct  orientation  of  geoid  and 
spheroid.  A  geodetic  latitude  (or  spheroidal  latitude  as  it 
should  perhaps  be  more  properly  called)  is  something  that 
cannot  be  directly  measured,  and  therefore  it  seems  that  the 
plumb-line  deviations  for  even  a  particular  spheroid  cannot 
be  absolutely  found  until  observations  have  been  made  over 
an  extent  of  country  wide  enough  to  enable  us  to  judge  of 
the  laws  governing  the  geoid  itself.  A  very  slight  change  in 
the  position  of  the  above  elliptical  arc  may  add  or  subtract  a 
constant  quantity  from  each  of  the  angles  between  the  true 
verticals  and  the  normals.  The  differences  of  the  plumb-line 
deflections  at  neighboring  stations  will,  however,  remain 
closely  the  same.  For  instance,  if  two  plumb-line  deflections 


86.  CONCLUSION.  245. 

are  3". 50  and  I ''.25  for  a  certain  spheroid,  another  spheroid 
may  be  drawn  making  them  2". 75  and  o".$o,  the  difference 
being  2". 2 5  in  both  cases.  ** 

The  above  figure  gives  a  very  exaggerated  picture  of  the 
relation  between  spheroid  and  geoid.  The  greatest  plumb- 
line  deflections  are  about  30",  and  it  is  unusual  to  find  them 
exceeding  15";  this  small  angle  could  not,  of  course,  be  seen 
in  a  common  drawing,  and  hence  in  any  true  representation 
the  spheroidal  and  geoidal  sections  should  be  drawn  parallel. 

It  is  further  to  be  noted  that  these  plumb-line  deflections 
in  longitude  as  well  as  in  latitude,  and  also  that  the  astronomi- 
cal and  geodetic  azimuths  of  a  line,  do  not  agree.  Thus  at 
Parkersburg  station,  on  the  U.  S.  Lake  Survey,  the  plumb- 
line  deflection  in  latitude  was  1^.47  toward  the  south,  the 
deflection  in  longitude  was  0^.70  toward  the  west,  and  the 
discrepancy  in  azimuth  was  o" .76.  The  greatest  deflection 
in  latitude  found  on  this  survey  was  io".77,  and  the  greatest 
one  in  longitude  12".  13,  while  the  greatest  discrepancy  in 
azimuth  was  ii'f.$6. 

86.  CONCLUSION. 

There  have  now  been  briefly  set  forth  a  history  of  geodesy 
and  the  elements  of  geodetic  theory  and  practice.  It  has 
been  seen  that  the  first  idea  of  the  shape  of  the  earth  was 
that  of  the  plane,  and  the  second  that  of  the  sphere.  Assum- 
ing it  to  be  a  sphere  measurements  and  computations  were 
made  to  find  its  size;  these  being  discordant  it  was  assumed 
to  be  a  spheroid  and  more  elaborate  investigations  were 
undertaken.  Assuming  it  to  be  an  ellipsoid  other  computa- 
tions were  made.  The  question  now  arises  whether  the  form 
of  the  geoid  can  be  deduced  so  that  definite  statements  can 
be  made  regarding  its  size  and  figure. 

Compared  with  a  spheroid  of  equal  volume,  the  geoid  has 
a  very  irregular  surface,  now  rising  above  that  of  the  spheroid, 


246  THE   FIGURE    OF  THE    EARTH.  X, 

now  falling  below  it,  and  ever  changing  the  law  of  its  curva- 
ture, so  as  to  conform  to  the  varying  intensity  and  direction 
of  the  forces  of  gravity.  Where  the  earth's  crust  is  of  greatest 
density  there  it  rises,  where  the  crust  is  of  least  density  and 
thickness  there  it  sinks.  From  a  scientific  point  of  view  it 
will  be  valuable  to  know  the  laws  governing  its  form  and  size; 
from  a  practical  point  of  view  it  appears  that  until  these  are 
known  the  earth's  figure  can  never  be  accurately  represented 
by  a  sphere  or  spheroid  or  ellipsoid,  or  other  geometrical 
form.  For  instance,  if  it  be  desired  to  represent  the  earth 
by  an  oblate  spheroid,  the  best  and  most  satisfactory  one 
must  be  that  having  an  equal  volume  with  the  geoid,  and 
whose  surface  everywhere  approaches  as  nearly  as  possible  to 
the  geoidal  surface.  Such  a  spheroid  cannot,  of  course,  be 
found  until  more  and  better  data  concerning  the  geoid  have 
accumulated,  yet  what  has  already  been  said  is  sufficient  to 
indicate  that  the  dimensions  at  present  used  are  probably 
somewhat  too  large.  Granting  that  in  general  the  geoid  rises 
above  this  spheroid  under  the  continents  and  falls  below  it  on 
the  seas  it  seems  evident,  since  the  area  of  the  oceans  is 
nearly  three  times  that  of  the  lands,  that  the  intersection  of 
the  two  surfaces  will  generally  be  some  distance  seaward  from 
the  coast  line,  as  seen  at  b  in  the  figure  of  the  last  Article. 
Now  geodetic  surveys  can  only  be  executed  on  the  continents, 
and  even  if  they  be  reduced  to  the  sea  level  at  the  coast,  or 
to  a  in  the  diagram,  the  elements  of  a  spheroid  deduced  from 
them  will  be  too  large  to  satisfy  the  above  condition  of 
equality  of  volumes,  for  the  ellipse  through  a  is  evidently 
larger  than  that  through  b.  At  present  it  would  be  purely  a 
guess  to  state  what  quantity  should  be  subtracted  from  the 
semi-axes  of  the  Clarke  spheroid  on  account  of  these  consid- 
erations. 

For  a  locality  where  precise  astronomical  and  geodetic  work 
has  been  done  a  fair  picture  of  the  relation  of  the  spheroid  to 
the  geoid  may  be  obtained.  But  on  the  oceans,  where  such 


86.  CONCLUSION.  247 

work  cannot  be  executed,  it  will  generally  be  impossible  to 
secure  numerical  comparisons.  The  word  Geoid,  in  fact, 
with  all  the  fruitful  ideas  Herein  implied,  is  comparatively 
new,  it  having  been  coined  in  1872.  Its  mathematical 
properties,  resulting  from  its  definition,  have  been  studied, 
and  Bruns  has  demonstrated  that  the  mathematical  figure  of 
the  earth  may  be  determined  independently  of  any  hypotheti- 
cal assumption,  provided  that  there  have  been  observed  at 
and  between  numerous  stations  five  classes  of  data,  namely, 
astronomical  determinations  of  latitude,  longitude,  and 
azimuth,  base  line  and  triangulation  measurements,  vertical 
angles  between  stations,  spirit  leveling  between  stations,  and 
determinations  of  the  intensity  of  the  forces  of  gravity. 
These  five  classes  are  sufficient  for  the  solution  of  the 
problem,  but  also  necessary;  that  is,  if  one  of  them  does  not 
exist,  a  hypothesis  must  be  made  concerning  the  shape  of  the 
earth's  figure.  These  complete  data  have,  however,  never 
yet  been  observed  for  even  an  extent  of  country  so  small  as 
England,  a  land  probably  more  thoroughly  surveyed  than  any 
other.  To  render  geodetic  results  of  the  greatest  scientific 
value,  it  is  hence  necessary  that  either  the  pendulum  or  some 
other  instrument  of  precision  should  be  employed  to  deter- 
mine the  relative  intensity  of  the  forces  of  gravity  at  the 
principal  triangulation  stations,  and  that  trigonometric  level- 
ing by  vertical  angles  should  be  brought  to  greater  perfection. 
These  conclusions  appear  to  neglect  the  circumstance  that 
the  geoid  is  not  a  fixed  and  invariable  figure.  Atmospheric 
influences  are  continually  at  work  to  tear  down  the  mountain 
ranges  and  fill  up  the  ocean  basins;  as  this  goes  on  the  geoid 
tends  to  become  more  uniform  in  curvature.  Internal  fires 
cause  parts  of  the  earth's  crust  to  slowly  rise  and  fall,  and 
immediately  the  geoidal  surface  undergoes  a  like  alteration. 
These  changes  are,  however,  probably  slight  compared  to 
those  caused  by  the  conical  rotation  of  the  earth's  axis  around 
its  mean  position  in  its  period  of  about  425  days.  Owing  to 


248  THE    FIGURE    OF    THE    EARTH.  X. 

this  rotation  all  astronomical  latitudes,  longitudes,  and  azi- 
muths are  subject  to  periodic  changes,  and  the  position  of 
the  geoid  with  respect  to  the  spheroid  is  constantly  varying. 
It  is  hence  plain  that  the  geoid  can  never  be  used  as  a  figure 
for  standard  reference  in  geodetic  surveying.  On  the  con- 
trary, a  spheroid  of  revolution,  or  an  ellipsoid  with  three 
unequal  axes,  must  always  be  employed  as  the  standard 
figure,  its  size  and  shape  being  so  determined  as  to  render  a 
minimum  the  sum  of  the  squares  of  all  the  plumb-line  deflec- 
tions that  occur  during  a  complete  cycle  of  the  conical  axial 
rotation. 

Since  the  first  edition  of  this  book  was  issued,  a  volume 
called  "  The  Transcontinental  Triangulation  and  the  Ameri- 
can Arc  of  the  Parallel,"  by  Charles  A.  Schott,  has  been 
published  by  the  United  States  Coast  and  Geodetic  Survey 
(Washington,  1900).  This  contains  interesting  conclusions 
regarding  the  surface  of  the  geoid  and  the  best  spheroid  to 
represent  it.  The  transcontinental  triangulation  extended 
from  Cape  May  on  the  Atlantic  coast  to  Point  Arena  on  the 
Pacific  coast,  the  observed  difference  in  longitude  between 
these  two  stations  being  48°  46'  oo".$8,  while  that  obtained  by 
computation  through  the  triangulation  was  48°  45'  35".93. 
The  computed  distance  between  these  stations  on  the  parallel 
of  39°  is  4224009.8  meters,  the  uncertainty  of  this  result 
being  Ts-nnnp  which  is  equivalent  to  0.38  inches  per  statute 
mile.  The  average  curvature  of  the  geoid  on  the  parallel  of 
39°  approaches  closely  that  of  the  Clarke  spheroid  of  1 866  for 
the  eastern  part  of  the  arc,  from  New  Jersey  to  Kansas,  while 
that  of  the  remainder  agrees  better  with  the  Bessel  spheroid. 
On  the  whole,  this  arc  across  the  American  continent  seems 
to  demand  an  intermediate  spheroid,  which,  in  latitude  39°, 
has  nearly  86  624  meters  as  the  length  of  one  degree  of  longi- 
tude, this  being  about  6  meters  less  than  the  length  given  by 
the  Clarke  spheroid  and  about  8  meters  more  than  the  length 
given  by  the  Bessel  spheroid. 


CONCLUSION. 


2480 


In  the  same  volume  Schott  gives  several  preliminary  com- 
binations of  American  geodetic  arcs  for  the  deduction  of  the 
elements  of  a  spheroid  whicjj  best  represents  the  surface  of 
the  geoid  on  the  western  continent.  The  following  shows 
the  results  obtained,  the  elements  of  the  Bessel  and  Clarke 

SPHEROIDS  FROM  AMERICAN  ARCS. 


IArcs  Combined. 

Semi-major 
Axis  a, 

Semi-minor 
Axis  b, 

Elliptic!  ty 

Meters. 

Meters. 

Intercontinental  arc  of  parallel  | 
Lake  Superior  arc  of  meridian  \ 

6377912 

6356309 

1/295.2 

Intercontinental  arc  of  parallel  ) 
Peruvian  arc  of  meridian            \ 

6378027 

6356819 

1/300.7 

Lake  Erie  arc  of  parallel            ) 
Peruvian  arc  of  meridian            f 

6379822 

6357716 

1/288.6 

Lake  Superior  arc  of  meridian  ) 
Peruvian  arc  of  meridian            J 

6  377  577 

6356577 

I/303-7 

Atlantic  arc  of  meridian              ) 
Peruvian  arc  of  meridian            f 

6378054 

6357175 

I/305-5 

Bessel's  spheroid  of  1841 

6377397 

6356079 

1/299.2 

Clarke's  spheroid  of  1866 

6378206 

6356584 

1/295.0 

spheroids  being  given  for  comparison  at  the  foot  of  the  table. 
All  these  combinations  demand  a  spheroid  somewhat  larger 
than  that  of  Bessel  in  order  to  fit  the  geoidal  surface  upon 
the  western  continent.  The  first  combination  in  the  table 
results  in  a-  spheroid  closely  agreeing  with  that  of  Clarke. 

It  is  seen  from  the  above  table  that  the  geodetic  work  done 
in  the  United  States  has  been  of  great  value  in  affording 
knowledge  regarding  the  figure  of  the  earth.  This  work  has 
occupied  more  than  half  a  century.  The  Atlantic  arc  above 
noted  consists  of  two  parts,  the  northern  or  Nantucket  arc 
being  measured  between  the  years  1845  and  1866,  while  the 
southern  or  Pamlico-Chesapeake  arc  was  measured  between 
1844  and  1876.  The  Lake  Superior  meridian  arc  and  the 
Lake  Erie  arc  of  the  parallel  were  completed  in  1882,  these 
being  the  work  of  the  Lake  Survey,  while  the  Atlantic  meridian 


248^  THE    FIGURE    OF    THE    EARTH.  X. 

arc  and  the  intercontinental  arc  of  the  parallel,  as  well  as 
others  not  here  considered,  have  been  done  by  the  Coast  and 
Geodetic  Survey.  The  Atlantic  arc  has  been  extended  south- 
ward to  the  Gulf  of  Mexico,  and  work  is  in  progress  upon  a 
longer  meridian  arc  extending  from  Nebraska  to  Texas. 

In  1902  a  volume  called  "Eastern  Oblique  Arc  of  the 
United  States  and  Osculating  Spheroid"  was  issued  by  the 
Coast  and  Geodetic  Survey  in  which  Schott  deduced  another 
spheroid.  This  arc  extends  from  Maine  to  Louisiana,  em- 
bracing 15°  13'  45". o  in  latitude  and  22°  47'  26".  5  in  longi- 
tude. The  spheroid  best  representing  the  region  covered  by 
this  arc  was  found  to  have  a  semi-major  axis  of  6378  157 
meters  and  a  semi-minor  axis  of  6  357210  meters,  the  ellip- 
ticity  being  1/304.5.  This  discussion  was  the  last  work  of 
Charles  Anthony  Schott,  who  died  in  1901  at  the  age  of 
seventy-five  years,  his  contributions  to  the  science  of  geodesy 
in  America  having  been  more  numerous  and  valuable  than 
those  of  any  other  man. 

Lastly  it  may  be  noted  that  in  1899  work  upon  a  remeas- 
urement  of  the  Peruvian  arc  was  begun  by  the  government  of 
France,  it  being  intended  to  extend  the  old  arc  one  degree 
northward  and  two  degrees  southward,  so  as  to  embrace  an 
amplitude  of  about  six  degrees.  When  this  important  work 
is  completed  there  will  be  data  at  hand  for  computing  the 
elements  of  a  spheroid  that  will  better  represent  the  mean 
geoidal  surface  of  the  earth  than  any  heretofore  determined. 


87.  EXPLANATION  OF  THE  TABLES.  249 


CHAPTER  XI. 
TABLES. 

-  .     87.  EXPLANATION  OF  THE  TABLES. 

The  following  tables  have  been  mostly  compiled  from  the 
extended  tabulations  given  in  Reports  of  the  U.  S.  Coast  and 
Geodetic  Survey  and  in  the  Smithsonian  Geographical  Tables. 
They  will  be  sufficient  for  the  solution  of  the  problems  given 
in  this  volume,  although  in  some  cases  the  larger  tables  will 
be  more  convenient  in  interpolation.  All  dimensions  of  the 
earth  are  for  the  Clarke  spheroid  of  1866. 

The  meter  used  in  these  tables  agrees  with  that  of  the 
Smithsonian  volume,  its  relation  to  the  yard  being  exactly 
expressed  by  the  fraction  |||^-  and  to  the  foot  by  ff^J;  thus, 
the  number  of  feet  in  a  meter  is  3.2808333 ;  the  meter  of  the 
tables  of  the  Coast  and  Geodetic  Survey  published  prior  to 
1897  is,  however,  3.2808693  feet,  this  being  the  value 
deduced  by  Clarke's  comparisons  of  1866. 

Table  I  contains  mean  values  of  the  correction  for  refrac- 
tion to  be  added  to  all  vertical  angles  taken  upon  celestial 
objects.  The  values  given  are  mean  ones,  the  word  mean 
implying  average  atmospheric  conditions.  Under  unusual 
extremes  of  temperature  and  barometric  pressure  the  actual 
refraction  may  be  greater  or  less  than  these  mean  values,  and 
in  precise  work  thermometer  and  barometer  readings  are  to 
be  taken.  For  the  class  of  work  described  in  Chapter  V  the 
mean  values  of  the  table  will  be  amply  sufficient.  Further,  it 
may  be  remembered  that  for  altitudes  greater  than  30°  the 
mean  refraction  is  expressed  by  57 
apparent  altitude. 


eotA,  wlrere~#~~is  the 


CIVIL  ENGINEERING 
I!,  oi  C. 

UiWARY 


250  TABLES. 

Tables  II  and  III  contain  lengths  of  arcs  of  the  meridian 
and  parallels  in  miles,  meters,  and  feet.  The  length  stated 
for  a  meridian  arc  is  the  length  of  an  arc  whose  middle  point 
has  that  latitude;  thus  the  length  of  one  degree  of  latitude 
at  latitude  40°  is  the  length  from  latitude  39°  30'  to  latitude 
40°  30'.  The  lengths  given  for  arcs  of  longitude  are  measured 
on  the  parallels  of  latitude.  The  manner  of  computing  these 
tables  is  explained  in  Art.  61. 

Table  IV  contains  logarithms  of  the  radii  of  curvature  of 
the  meridian  and  of  the  prime-vertical  normal  section;  the 
derivation  of  these  values  is  explained  in  Art.  62.  The  addi- 
tion of  the  two  logarithms  gives  the  logarithm  of  the  square 
of  the  radius  of  the  osculatory  sphere,  since  R^R^  =  R*.  The 
fourth  column  contains  logarithms  of  the  radii  of  curvature 
of  the  parallels,  computed  by  the  formula  r  =  R^  cosL.  The 
last  column  gives  values  of  the  logarithm  of  206  265/2^^,, 
which  is  to  be  added  to  the  logarithm  of  tw.'ce  an  area,  in 
meters,  in  order  to  obtain  the  logarithm  of  the  spherical  ex- 
cess of  the  area,  as  explained  in  Art.  63. 

Table  V  contains  logarithms  of  the  constants  to  be  used  in 
computing  geodetic  latitudes,  longitudes,  and  azimuths.  The 
expressions  for  these  constants  and  the  manner  of  obtaining 
their  values  are  given  in  Art.  64.  The  factor  £,  given  in  the 
last  column,  will  be  needed  in  computing  large  triangles  and 
is  hence  retained  in  the  table,  although  its  derivation  has  not 
been  explained  in  this  volume. 

Table  VI  contains  constants  and  their  logarithms  which 
will  be  of  service  in  many  computations.  It  will  rarely  be 
necessary  to  use  more  than  seven  of  the  nine  decimals. 

It  may  be  noted  that  in  these  tables  and  throughout  this 
book  the  logarithms  of  numbers  less  than  unity  are  written 
with  a  negative  characteristic;  thus,  log  sin  o°  01'  is  2.24903, 
not  8.24903  as  given  in  most  tables.  When  using  logarithmic 
t  ibles  the  student  should  note  that  8  and  9  mean  $  and  1  and 
should  write  them  thus  in  his  computations. 


TABLES. 
TABLE  I.     MEAN  CELESTIAL  REFRACTION. 


251 


Apparent 
Altitude. 

Refrac- 
tion. 

Apparen 
Altitude 

Refrac- 
tion. 

Apparent 
Altitude. 

Refrac- 
tion. 

Apparen 
Altitude 

Refrac- 
tion. 

0  00 

34  54-1 

8   00 

6  29.6 

23  oo 

2  15.2 

43 

I  01.8 

10 

32  49-2 

20 

15-2 

30 

12.0 

44 

o  59-7 

2O 
30 
40 

30  52-3 
29  03.5 
27  22.5 

40 

9  oo 
20 

01.8 

5  49-3 
37.6 

24  oo 

30 

2  08.9 
O6.O 

45 
46 
47 

57.7 
55.7 

53-8 

50 

25  49-8 

4° 

26.5 

25  oo 

2  03.2 

48 

51-9 

I   00 
IO 

24  24.6 
23  06.7 

IO   OO 

20 

5  16.2 

06  4 

30 

26  oo 

00-5 

i  57-8 

49 
50 

50.2 
o  48.4 

2O 
30 
40 
50 

21  55.6 
20  50.9 

19  51-9 
18  58.0 

40 

II   00 
20 

v  .  44. 

4  57.2 

4  48.5 
40.2 

30 

27  oo 

30 

55-3 
i  52.8 
50.5 

5i 

52 
53 
54 

46.7 
45.1 
43.5 
41-9 

2   OO 

18  08.6 

40 

32.4 

28  oo 

i  48.2 

55 

40.4 

10 

17  23.0 

12   OO 

4  25.0 

30 

46.0 

56 

38.9, 

20 
30 
40 
50 

16  40.  7 
16  00.9 
15  23.4 
14  47-8 

2O 

40 

13  oo 
20 

18.0 
n-3 
4  04.9 

3  58.8 

29  oo 

30 

30  oo 

30 

i  43-8 
41.7 

i  39-7 
37-7 

57 
58 
59 

60 
fr 

37-5 
36.1 
34.7 

o  33-3 

3  oo 

IO 
20 

14  14.6 
13  43-7 

T  C   O 

40 

14  oo 

53-o 
3  47-4 

31  oo 

30 

i  35-8 
33-9 

OI 

62 
63 

32.0 
30-7 
29.4 

30 
40 

50 
4  oo 

IO 
20 

30 

1  3  .  u 

12  48-3 
23-7 
00-7 

II  38.9 
I8.3 

10  58.6 
39-6 

20 
40 

15  oo 

20 

40 

16  oo 
20 

42.1 

37-0 

3  32.1 
27.4 
22.9 

3  18.6 
14-5 

32   00 

30 

33  oo 

30 

34  oo 
30 

i  32.1 
30.3 
i  28.7 
27.0 

i  25.4 
23.8 

64 
65 
66 

67 
68 
69 

70 

28.2 
26.9 
25-7 
24.5 
23-3 

22.2 
O  21.  O 

40 

21.2 

40 

10.5 

35  oo 

i  22.3 

7i 

72 

19.9 

18.8 

50 

°3-3 

17  oo 

3  06.6 

30 

20.8 

/• 

73 

17.7 

5  oo 

9  46.5 

20 

02.9 

36  oo 

I  19-3 

74 

16.6 

IO 

30-9 

40 

2  59-3 

30 

17.8 

75 

15-5 

20 
30 
40 

16  o 
01.9 

8  48.4 

18  oo 
20 

2  55-8 
52-5 

37  oo 
30 

I  16.5 

15.1 

76 
77 

78 

14-5 
13-4 
12.3 

50 

35-6 

4° 

49-3 

38  oo 

I  13.8 

79 

II.  2 

6  oo 

8  23.3 

19  oo 

2  46.  i 

30 

12.4 

80 

0  10.2 

IO 

11.  6 

20 

43-i 

39  oo 

I  II.  2 

8l 

Og.I 

20 

00.3 

40 

40.2 

30 

09.9 

82 

08.  1 

30 
40 

7  49-5 
39-2 

20  oo 
20 

2  37-3 
34-5 

40  oo 

I  08.7 

83 
84 

07.1 
06.  1 

50 

29.2 

40 

3i-9 

30 

07.  5 

85  - 

05.1 

7  oo 

10 

7  19-7 
10.5 

21   00 
20 

2  29.3 

26.8 

41  oo 

30 

i  06.3 
05.1 

86 
87 

04.1 
03.1 

88 

O2  .  1 

20 

01.7 

40 

24-3 

42  oo 

i  04.0 

80 

OI  .  I 

30 

6  53-3 

22   00 

2  21.9 

30 

02.9 

°v 

40 

45  •  i 

50 

37.  21 

30 

18.5 

TABLES. 


TABLE  II.     LENGTHS  OF  ARCS  OF  THE  MERIDIAN. 


Latitude. 

One  Degree. 

One  Minute. 
Meters. 

One  Second. 

Miles. 

Meters. 

Meters. 

Feet. 

O° 

68  .  703 

110568 

1842.81 

30.713 

100.766 

5 

.709 

577 

2-95 

.716 

•773 

10 

•725  : 

602 

3-37 

•723 

•797 

15 

.751  - 

644 

4.06 

•734 

•834 

20 

68.786 

110700 

1844.98 

30.750 

100.886 

21 

•794 

713 

5-21 

•737 

.897 

22 

.802 

726 

5-44 

.741 

.910 

23 

.810 

740 

5-68 

.761 

.922 

24 

.819 

754 

5-91 

.765 

•935 

25 

68.829 

no  769 

1846.15 

30.769 

100.949 

26 

.838 

•785 

6.41 

•773 

.963 

27 

.848 

800 

6.67 

.778 

•977 

28 

.858 

816 

6.94 

.782 

•992 

29 

.868 

833 

7.21 

.787 

101.007 

30 

68.879 

110850 

1847.49 

30.791 

101.022 

31 

.889 

867 

7-78 

.796 

.038 

32 

.900 

884 

8.07 

.801 

-054 

33 

.911 

902 

8-37 

.806 

.070 

34 

•923 

920 

8.67 

.811 

.086 

35 

68.934 

110939 

1848.98 

30.816 

IOI.IO3 

36 

.946 

957 

1849.29 

.821 

.120 

37 

•957 

976 

9.60 

.827 

•137 

38 

.969 

995 

9.92 

.832 

•155 

39 

.981 

in  014 

1850.24 

•837 

.172 

40 

68.993 

111034 

1850.56 

30.843 

IOI.I90 

41 

69.005 

053 

0.89 

.848 

.208 

42 

.017 

073 

1.22 

.854 

.225 

43 

.029 

093 

1-54 

.859 

•243 

44 

.042 

112 

1.87 

.865 

.26l 

45 

69.054 

III  132 

1852.20 

30.870 

IOI.279 

46 

.067 

152 

2.53 

.876 

.297 

47 

.079 

172 

2.86 

.881 

•315 

48 

.091 

191 

3-19 

.•886 

•333 

49 

.103 

211 

3-51 

.892 

•351 

50 

69.115 

III  231 

1853-84 

30.897 

101.369 

51 

.127 

250 

4.  16 

•903 

.387 

52 

•139 

269 

4.49 

.908 

•404 

53 

•  151 

288 

4.81 

•913 

.422 

54 

.163 

307 

5-12 

.919 

-439 

55 

69-175 

III  326 

1855.43 

30.924 

101.456 

60 

.230 

4l6 

5-69 

•949 

.538 

65 

.281 

497 

5.82 

.971 

.612 

70 

•324 

567 

5-95 

.991 

.676 

75 

69.360 

in  624 

1860.40 

31-007 

101  .728 

80 

.386 

666 

I.  IO 

.018 

.766 

85 

.402 

692 

i-53 

.026 

.788 

90 

.407 

701 

1.68 

.028 

•  797 

TABLES. 


253 


TABLE  III.     LENGTHS  OF  ARCS  OF  PARALLELS. 


Latitude. 

One  Degree. 

One  Minute. 
j£.    Meters. 

One  Second. 

Miles. 

Meters. 

Meters. 

Feet. 

0° 

69.171 

Ill  322 

1855.36 

30.923 

101.452 

5 

68.911 

ioi  901 

1848.35 

30.bo6 

101.069 

10 

68.128 

109  642 

1827.36 

30.456 

99.921 

15 

66.830 

107  553 

I792-55 

29.709 

98.018 

20 

65.026 

104  650 

1744.16 

29.069 

95-372 

21 

64  .  606 

103  973 

1732.89 

28.881 

94-755 

22 

64.166 

103  265 

1721.08 

28.685 

94.  no 

23 

63  .  706 

102  525 

1708.76 

28.479 

93-436 

24 

63.227 

ioi  755 

1695.91 

28.265 

92.733 

25 

62.729 

100953 

1682.55 

28.042 

92.003 

26 

62.212 

IOO  121 

1668.68 

27.811 

91.244 

27 

61.676 

99258 

1654.30 

27.562 

90.458 

28 

6l  .  121 

98365 

1639.41 

27.322 

89  .  644 

29 

60.548 

97442 

1624.03 

27.067 

88.803 

30 

59.956 

96489 

1608.  16 

26.803 

87-935 

31 

59.345 

95  507 

I59L79 

26.530 

87.040 

32                   58.717 

94496 

1574-94 

26.  249 

86.118 

33              58.071 

93456 

i557.6i 

25.960 

85.171 

34 

57.407 

92388 

I539.8o 

25.663 

84.197 

35 

56.726 

91  291 

1521.52 

25.359 

83.198 

36 

56.027 

90  167 

1502.78 

25.046 

82.1-73 

37 

55-3" 

89  015 

1483-58 

24.726 

81.123 

38 

54-578 

87836 

1463-93 

24-395 

80.048 

39, 

53.829 

86630 

I443.83 

24  .  064 

78.949 

40 

53-063 

85937 

1432.28 

23.871^ 

77.826 

41 

52.281 

84138 

1402.31 

23.372 

76.679 

42 

51.483 

82854 

1380.90 

23  015 

75.508 

43 

50.669 

81  544 

1359-07 

22.651 

74-315 

44 

49  .  840 

80  209 

1336.82 

22.28O 

73.098 

45 

48.995 

78  850 

I3I4-I7 

21.903 

71-859 

46 

48.135 

77466 

1291.  ii 

21.518 

70.599 

47 

47.261 

76059 

1267.65 

21.128 

69.316 

48 

46.372 

74629 

1243.81 

20.730 

68.012 

49 

45.469 

73  175 

1219.58 

20.362 

66.687 

50 

44-552 

71699 

1194-65 

19.911 

65.342 

43.621 

70201 

1170.01 

19.500 

63.977 

52 

42.676 

68681 

.     1144.68 

19.078 

62.  592 

53 

41.719 

67  140 

1119.01 

18.650 

61.188 

54 

40.749 

65  579 

1092.98 

18.216 

59.765 

55 

39.766 

63997 

1066.62 

17-777 

58.323 

60 

34.674 

55803 

930.05 

15.501 

50.855 

65 

29-3I5 

47178 

786.30 

13.105 

42.995 

7  ;) 

23.729 

38  189 

636.48 

10.608 

34-803 

75 

17.960 

28  904 

481.73 

8.029 

26.341 

80 

12.051 

19395 

323-24 

5.387 

17-675 

85 

6.049 

9735 

162.25 

2.704 

8.871 

90 

0 

o 

o 

0 

0 

254 


TABLES. 


TABLE  IV.     LOGARITHMS  FOR  GEODETIC  COMPUTATIONS. 

IN    METERS. 


Latitude. 

Radius  of  Curvature. 

Factor  m  for 
Spherical  Excess. 

Of  Meridian, 
T?I. 

Of  Prime  Vertical 
Normal  Section,  A"2. 

Of  Parallel, 
r. 

0° 

6.801  7538 

6.8047034 

6.8047034 

9  .  40  694 

5 

7873 

7137 

.8030579 

689 

10 

8868 

7478 

.7980993 

676 

15 

6.802  0492 

8019 

.7897457 

664 

20 

6.8022698 

6.8048754 

6.777  86l2 

9.4O  624 

21 

3204 

8922 

.7750439 

618 

22 

3729 

9098 

.7720757 

611 

23 

4274 

9279 

.  768  954O 

604 

24 

4838 

9467 

.765  6769 

596 

25 

6.802  5418 

6.8049661 

6.762  2418 

9.40  589 

26 

6016 

9860 

.7586462 

582 

27 

6633 

6.805  0065 

.7548874 

571 

28 

7264 

0275 

.7509624 

565 

29 

7918 

0492 

.746  8685 

555 

30 

6.802  8572 

6.805  0712 

6.7426018 

9.40  546 

31 

9246 

0938 

.738  1594 

536 

32 

9933 

1176 

.7335382 

527 

33 

6.8030631 

1399 

.7287313 

5i8 

34 

1341 

I635 

.7237377 

509 

35 

6.803  2062 

6.805  1876 

6.718  5611 

9.40  500 

36 

2791 

2118 

.713  1694 

49° 

37__ 

3528 

2364 

.7075850 

480 

38 

4273 

2613  ••- 

•  7oi  7934 

470 

39 

5025 

2863 

.695  7889 

460 

40 

6.803  5782 

6.805  3Il6 

6.689  5656 

§.40450 

4i 

6545 

3369 

.683  1168 

440 

42 

73ii 

3625 

.6764350 

430 

43 

8080 

3880 

•6695155 

419 

44 

8851 

4138 

.  662  3479 

409 

45 

6.803  9623 

6.8054395 

6.  653  65*75 

9.40399 

46 

6.8040395 

4652 

.6472365 

389 

47 

1166 

4908 

.6392741 

379 

48 

1935 

5166 

•  631  0275 

368 

49* 

2705 

5422 

.6224851 

358 

50 

6.8043466 

6.8055677 

6.613  7252 

9.40348 

5i 

4224 

5929 

.  604  4647 

338 

52 

"  4976 

6l8o 

.  594  9600 

328 

53 

5723 

6429 

.585  1059 

3i8 

54 

7195 

6676 

.5748863 

301 

55 

6.8048626 

6.805  6919 

6.564  2832 

§.40284 

60 

6.805  0693 

8086 

.5047786 

252 

65 

3857 

9141 

.431  8624 

210 

70 

J  659i 

6.8060052 

.3400569 

173 

75 

6.805  8809 

6.806  0791 

6.2190753 

§.40144 

80 

6  .  806  0443 

1336 

.045  8038 

125 

85 

1444 

1670 

5  .  746  4630 

1  08 

90 

1782 

1782 

00 

104 

TABLES. 


255 


TABLE  V.     LOGARITHMS  FOR  THE  LMZ  PROBLEM. 

IN    METERS. 


Latitude. 

A' 

B 

*   c 

D 

E 

O° 

2.5099613 

2.5126713 

—   CO 

—  CO 

15.6124 

5 

7114 

6378 

10.34885 

9.6275 

.6223 

10 

6773 

5383 

.65308 

..9220 

.6511 

^5 

6232 

»   3759 

.  38460 

8.0871 

.6970 

20 

2.5095497 

2.5I2I553 

10.96732 

8.1964 

15-7574 

21 

5329 

1047 

,.99036 

.2138 

.7712 

22 

5153 

0522 

9.01252 

.2302 

.7852 

23 

4972 

2.5119977 

.03389 

.2487 

-7997 

24 

4784 

9613 

•05455 

.2629 

.8146 

25 

2.5094592 

2.511  8834 

§.07456 

8.2762 

15.8300 

26 

4391 

8231 

.09399 

.2885 

.8458 

27 

4186 

7619 

.11289 

.3000 

.8620 

28 

3976 

6988 

-I3I3I 

.3107 

•8785 

29 

3760 

6341 

•  J493i 

.3206 

•8955 

30 

2.5093540 

2.511  5681 

9.16691 

8.3298 

15.9127 

31 

3315 

5006 

.18415 

•3382 

•9304 

32 

3086 

4320 

.20107 

.3460 

.9484 

33 

2853 

3621 

.21771 

.3532 

.9667 

34 

2617 

2911 

.  23408 

•3597 

•9853 

35 

2.5092377 

2.511  2191 

9.25023 

8.3656 

14.0043 

36 

2134 

1462 

.26616 

•3709 

.0237 

37 

1888 

0724 

.28192 

•3756 

•0433 

38 

1639 

2.5109979 

.29752 

•3797 

.0633 

39 

1389 

9227 

.31298 

•3833 

.0836 

40 

2.509  1136 

2.5108469 

§.32833 

8.3863 

14.1043 

41 

0882 

7707 

•34357 

.3888 

-1253 

42 

0627 

6941 

.35874 

.3907 

.1467 

43 

0471 

6172 

.37385 

.3921 

.1684 

44 

0114 

540i 

•38893 

•3929 

.1905 

45 

2.5089856 

2.5104629 

§.40399 

8-3933 

14.2130 

46 

9599 

3857 

.41905 

•3932 

•2359 

47 

9342 

3086 

•43413 

•3924 

.2592 

48 

9065 

2316 

.44925 

.3912 

.2830 

49 

8830 

1550 

.  46442 

•3894 

.3071 

50 

2.5088575 

2.5100787 

§.47967 

8.3871 

14.3318 

51 

8323 

0028 

.49501 

.3842 

•3569 

52 

8072 

2.5099275 

.51047 

.3808 

.3826 

53 

7823 

8529 

.52607 

•3768 

.4088 

54 

7576 

7790 

.54182 

.3722 

•4355 

55 

2.5087333 

2.5097059 

9.55776 

8.3671 

14.4629 

60 

6166 

3559 

.64108 

•  3320 

.6102 

65 

Sin 

0394 

•73342 

.2790 

.7802 

70 

4199 

2.508  7660 

.84066 

.1998 

•  9836 

75 

2.5083460 

2.508  5442 

9.97338 

8.0909 

18.2410 

80 

2915 

3808 

8-15493 

9.9262 

_.9986 

85 

2581 

2807 

.45913 

.6319 

12.2038 

9° 

2469 

2469 

+  00 

—  00 

+  «> 

256 


TABLES. 


TABLE  VI.     CONSTANTS  AND  THEIR  LOGARITHMS. 


Name. 
(Radius  of  circle  or  sphere  =  i.) 

Symbol. 

Number. 

Logarithm. 

Area  of  circle 

it 

3    141  592654 

0.497  149873 

Circumference  of  circle 

2ir 

6.283  185  307 

0.798  179868 

Surface  of  sphere 

471 

12.566  370614 

1.099  209864 

I* 

0.523  59s  776 

1.7l8  998  622 

Quadrant  of  circle 

0.785  398  163 

1.895089881 

Area  of  semicircle 

\" 

1.570796327 

o.  196  119  877 

Volume  of  sphere 

4.187  790205 

0.622088  609 

7t* 

9.869  604401 

0.994299745 

n* 

1.772453851 

0.248  574936 

Degrees  in  a  radian 

I80/7T 

57-295  779513 

1.758  122632 

Minutes  in  a  radian 

IO8OO/7T 

3437.746771 

3.536273883 

Seconds  in  a  radian 

648ooo/7T 

2O6  264.806 

5.314425  133 

I/^ 

0.318  309  886 

1.502  850  127 

„ 

0.564  189  584 

1.751  425064 

i/*2 

o.ioi  321  184 

1.005  700255 

Circumference/36o 

arc 

0.017453293 

2.241  877  368 

sin 

0.017452406 

2.241855318 

Circumference/2  1600 

arc 

o.ooo  290  888 

4.463  726  117 

sin 

o.ooo  290  888 

4.463  726  III 

Circumference/I296ooo 

arc     ' 

0.000004848 

6.685  574867 

sin 

o  .  ooo  004  848 

6.685  574867 

Ba.se  Naperian  system  of  logs 

e 

2.718  281  828 

0.434  294  482 

Modulus  common  system  of  logs 

M 

0.434294482 

1.637  784311 

Naperian  log  of  10 

i/M 

2.302  585  093 

0.362  215  689 

hr 

0.4769363 

1.6784604 

Probable  error  constant 

hr  yT 

0.6744897 

1.8289754 

Feet  in  one  meter 

m/ft. 

3.2808333 

0.5159841 

Miles  in  one  kilometer 

km/mi. 

0.621  369  9 

1-7933502 

INDEX. 


257 


INDEX. 


Accidental  errors,  8,  9 
Adjustment  of  angles,  18,  43,  209,  211 
observations,  25,  219 
triangles,  17,  51,  213 
Adjustments,  complex,  229 
figure,  216 
levels,  94 
polygon,  222 
quadrilateral,  225 
station,  18,  51,  209 
Alignment  curve,  174 
Almanac,  nautical,  109 
Anaximander,  138 
Angles,  18,  28,  39,  206 
and  bases,  74 
7at  stations,  18,  51,  60,  209 
horizontal,  39,  206 
geodetic,  215 
on  sphere,  151 
•vertical,  33,  101,  104,  105 
Archimedes,  138 
Arcs  on  meridian,  160,  170,  252 

parallel,  171,  253 
Areas,  error  in,  32 

spherical,  151 
spheroidal,  175 
Aristotle,  138 

Arithmetic  mean,  7,  13,  27,  34 
Astronomical  latitude,  108,  166,  244 
notions,  108 
work,  107-135 
Atmosphere,  98,  249 
Axis  of  earth,  136,  169,  247 
Azimuth,  astronomical,  107,  109,  134 
by  altitude  of  sun,  114 


Azimuth,  by  Polaris,  117,  121 
by  solar  transit,  no 
geodetic,  36,  177 
on  sphere,  150,  153 
on  spheroid,  177,  215 
plane,  36,  55,  69 

Base  lines,  71-86 

Bars,  measuring,  71 

Bessel's  spheroid,  168 

Bonne's  projection,  190 

Boundary  lines,  230 

British  Ordinance  Survey,  168,  237 

Broken  bases.  83 

Bruns,  247 

Cassini,  142 

Celestial  refraction,  251 
Centrifugal  force,  233 
Central  standard  time,  129 
Circles,  148,  256 
Circuit  of  levels,  94 
Chain  triangle  nets,  67 
Clairaut's  theorem,  232 
Clarke's  ellipsoids,  238 

spheroid,  168,  237 
Coefficient  of  expansion,  76 

refraction,  102 

stretch,  76 
Columbus,  141 

Conditional  equations,  59,  218 
Conditioned  observations,  17,  59,219 
Constant  errors,  7 
Constants,  table  of,  256 
Coordinates,  36,  55,  69,  181,  194 


258 


INDEX. 


Coordinate  systems,  36,  182,  194 
Correlates,  method  of,  60,  220 
Cost  of  towers,  201 
Culmination  of  Polaris,  127 

moon,  133 
Curvature  of  earth,  88,  99 

radii  of,  157,  173,  254 
Curve  of  probability  of  error,  n,  26 

Declination  of  Polaris,  118 

sun,  in 

Deflection  of  plumb-line,  164,  243 
Degree  of  meridian,  141,  147,181,252 

parallel,  149,  171,  253 
Degree,  trapezoidal,  153,  176 
Delambre,  163 
Direct  observations,  17,  59 
Direction-angle,  195 
Direction  theodolite,  39,  207 
Directions,  method  of,  209,  210 
Dumpy  level,  97 
Duplicate  level  lines,  89 

Earth,  figure  of,  137-149,  232-248 
orbit  of,  240 
radius  o.f,  148 
radii  of  curvature,  173,  254 
Eastern  standard  time,  129 
Eccentric  signals,  205 
stations,  202 
Elements  of  ellipsoid,  238 

spheroid,  159,  i66t  168 
Ellipse,  equation  of,  157 
properties  of,  156 
radii  of  curvature,  158 
Ellipsoid,  earth  as,  235,  238 
Ellipticity,  156,  168,  232 
Elongation  of  Polaris,  118 
End  measures,  71 
Engineers'  level,  97 

transit,  114 
Equations,  normal,  20,  22 

observation,  16,  19,  22 
side,  61,  218 
Eratosthenes,  138 


Error,  law  of  probability,  9 

probable,  26 
Errors,  accidental,  8 
constant,  7 
residual,  8 
Errors  in  angles,  40 

base  lines,  78,  82 
triangles,  49 

Eccentricity  of  ellipse,  156, 
Excess,  spherical,   151,  156,  176,  213 
Expansion,  coefficient  of,  76,  78 

Fernel,  141 

Field  notes,  41,  44,  80 

Figure  adjustment,  60,  216 

Figure  of  earth,  137,  140,  147,  232-248 

Flamsted's  projection,  189 

Functions  of  observations,  32 

Gauss,  25 
Geodesy,  137-256 
Geodesic  lines,  174 
Geodetic  bases,  75 

coordinates,  181 

latitude,  183,  245 

leveling,  87,  97 

projections,  181,  189 

tables,  194,  249 

triangulation,  50,68,85,105, 

198-231 

Geography,  36,  108,  181,  240 
Geographical  tables,  176,  194 
Geoid,  242,  243,  245,  247 
Gore,  35 
Greeks,  speculations  of,  138 

Heights,   determination  of,  87,  103, 

105 

Heliotrope,  202,  204 
Hilgard,  180 
History    of    geodesy,   137-144,    160, 

232,  242 
Horizon,  108 

Horizontal  angles,  39,  44,  109,  206 
Hour  angle,  109 


INDEX. 


259 


Independent  angles,  42,  210 

observations,  17 

Indirect  observations,  17,  19,  22]  29 
Inverse  LMZ  problem,  186 

Jordan,  197 

Latitude,  36,  108,  165 

astronomical,  107 
by  a  star,  127 
by  the  sun,  126 
ellipsoidal,  236 
geodetic,  157,  183,  245 
spherical,  153 
spheroidal,  177 
variation  of,  136 
Lamp  signals,  205 
Lapland  arc,  144,  160 
Laplace,  161,  233 
Law  of  error,  9,  u 
Least  squares,  7-35 
history  of,  163 
literature  of,  34 
principle  of,  12,  15 
theory  of,  33 
Legendre,  34,  163 

Lengths  of  arcs  of  meridian,  170,  252 
parallel,  171,  253 
quadrant,  164,169 
Level  surface,  87,  100 
Leveling,  16,  87-106 
Linear  coordinates,  37,  194 

measurements,  26,  71 
Literature  of  geodesy,  35 
LMZ  computations,  183,  186,  255 

constants,  180,  255 
Logarithms  for   geodetic   computa- 
tions, 254,  255 

Logarithms  of  constants,  254-256 
Longitude,  36,  132,  135,  153,  177 
Loxodrome,  191 
Lunar  measurements,  132,  133 

Magellan,  141 
Mandeville,  140 


Map  projections,  189 
Mason  and  Dixon,  145 
Measurement  of  angles,  39,  44 
lines,  79 
meridian  arcs,  139, 

144 

with  tape,  79 

Mercator's  projection,  191 
Meridian  arcs,  139,  144,  147,  158, 160. 

167,  168,  170,  252 
Metallic  bars,  71 
tapes,  76 

Meter,  163,  169,  180 
Method  of  least  squares,  7-35,  l6s 
Mistakes,  8 
Monuments,  200 
Mosman,  202 
Most  probable  value,  8,  13,  15,  33^ 

N- point  problem,  67 
Nautical  almanac,  109 
Navigation,  126,  191 
Net  of  level  lines,  95 

triangles,  67,  109,  217,  229 
Newton,  142,  143,  232 
New  York  state  survey,  200 
Normal  equations,  20,  22,  24 

sections,  172 
Norwood,  141 

Oblate  spheroid,  143 

Observation  equations,  16,  19 

Observations,  conditional,  17 
direct,  17 
independent,  17 
indirect,  17-22 
weighted,  14 

Ocean  level,  reduction  to,  84 
surface,  138,  241,  242 

Orbit  of  earth,  240 

Orthographic  projections,  150,  189 

Osculatory  sphere,  175 

Ovaloid,  239 

Parallax  of  sun,  116 


260 


INDEX. 


Parallel  of  latitude,  149 

lengths  of,  171,  234,  253 
Pendulum  observations,  232,  241 
Pentagon,  60 
Peruvian  arc,  144 
Photographs,  198 
Picard,  141 
Plane  triangles,  54 

triangulation,  36-70 
Plato,  138 

Plumb-line  deflections,  164 
Polaris,  azimuth  by,  117.  121 
declination  of,  118 
latitude  by,  127 
magnitude  of,  123 
Posidonias,  140 
Polyconic  projection,  192 
Precise  astronomical  work,  134 
leveling,  97 
triangulation,  36 

Precision  of  measurements,  12,  25 
Primary  triangulation,  206,  218 
Prime  vertical,  132,  173 
Probable  error,  25 
of  angles,  43,  206 

azimuth,  113,  116 

bases,  75 

conditioned  observations,  222 

computed  values,  31 

level  lines,  91,  93,  96 

linear  measures,  72 

indirect  observations,  29 
Probability  curve,  n,  13 
Projections,  181,  189 
Prolate  spheroid,  143 

Quadrant  of  ellipsoid,  238 

spheroid,  163,  164 
Quadrilateral,  67,  152,  225 

Radian,  33,  256 

Radii  of  curvature,  85,  173,  175,  254 

Radius  of  earth,  148 

Reconnaissance,  198 

Rectangular  coordinates,  37,  183,  193 


Refraction,  celestial,   112,    115,    199, 

249,  251 

terrestrial,  8,  88,  99,  IOI 
Rejection  of  observations,  44 
Repeating  theodolite,  206 
Repetition  of  angles,  41 
Residual  errors,  8,  19,  30 

Sag  of  tape,  77 
Schott,  C.  A.,  248,  249^ 
Secondary  stations,  61,  64,  202 

triangulation,  53,  64,  202 
Semi-duplicate  level  lines,  89 
Sextant,  125,  128,  130 
Shooting  at  target,  10 
Side  equations,  61,  68 
Signals,  203 

Slope,  correction  for,  78 
Smithsonian  tables,  176,  249    • 
Snellius,  141 
Solar  transit,  no 
Soldner,  196 

Solution  of  equations,  24 
Sphere,  lines  on,  149 
areas  on,  151 
Spherical  angles,  151,  212 

coordinates,  154,  194 
excess,  152,  213 
geodesy,  137-155 

Spheroid,  dimensions  of,  168,  232,246 
oblate,  143 
prolate,  143 

Spheroidal  geodesy,  156-180 
Spirit  leveling,  87,  94 
Standard  time,  129 
Star,  azimuth  by,  117 
latitude  by,  116 
Station,  adjustment   at,  18,  42,  46, 

209,  2ii 
Stations,  200 

Sun,  azimuth  by,  no,  114 
declination  of,  ill 
latitude  by,  124 
Systematic  errors,  7,  40 

Tables,  249-256 


INDEX. 


26l 


Tapes,  standard,  71,  76 
Target  shooting,  9 
Temperature,  7,  71,  76,  78      .    s 
Time,  determination  of,  169 

standard,  161 
Theodolites,  206,  207 
Three-point  problem,  64 
Todhunter,  234 
Toise,  142 
Topography,  198 
Towers,  200,  202 
Transit,  engineers',  39 
Trapezoidal  degree,  176 
Triangle  adjustment,  17,  21,  49,  51 

computation,  53,57,212,215 

net,  67 

on  sphere,  151 

spheroid,  175 

Triangulation,  geodetic,  198-231 
plane,  3^-70,  107 

Uncertainty  of  a  base,  72,  74 

line,  50 
U.  S.  Bureau  of  Equipment,  109 

Weights  and  Meas- 
ures, 76 


U.  S.  Coa-st  and  Geodetic  Survey, 
75,  84,  102,  180,  194,  199,  202, 
231,  234,  249 

Lake  Survey,  18,  245 

Northern  Boundary  Survey, 
231 

Variations  in  geoid,  247 

latitude,  136 

of  ocean  surface.  241 

Verniers,  40 

Vertical  angles,  33,  101,  230 

Weighted  mean,  28 

observations,  14 
Weights,  19,  22,  25 

of  angles, ,43,  45.  214 
levels,  91 
lines,  73 
Wilson,  99 
Woodward,  82,  176,  194 

Yard,  169,  249 

Zenith  telescope,  135 
Zone,  spherical,  152 
spheroidal,  176 


CIVIL  ra.\EE8LNG 
(I  of  t 


SHORT-TITLE     CATALOGUE 

OF  THE 

PUBLICATIONS 

OF 

JOHN   WILtEY   &    SONS, 

NEW  YORK, 
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*  Abegg's  Theory  of  Electrolytic  Dissociation.    (Von  Ende.) i2mo,  i   25 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  i  25 

Alexeyeff's  General  Principles  of  Organic  Synthesis.     (Matthews.) 8vo,  3  oo 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo- 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Beard's  Mine  Gases  and  Explosions.     (In  Press.) 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose  ' 

Molecule i2mo,  2  50 

Bolduan's  Immune  Sera 12mo,  i  50? 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  50 

Brush  and  Penfield's  Manual  of  Determinative  Mineralogy 8vo,  4  oo 

*  Claassen's  Beet-sugar  Manufacture.     (Hall  and  Rolfe.) 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Boltwood.).  .8vo,  3  oo 

Cohn's  Indicators  and  Test-papers i2mo,  2  oo 

Tests  and  Reagents 8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.   (Schaeffer.).  .  .  i2mo,  i  50 

*  Danneel's  Electrochemistry.     (Merriam.) i2mo,  i  25 

Dolezalek's   Theory  of   the   Lead  Accumulator   (Storage   Battery).         (Von 

Ende.) i2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  i   25 

*  Fischer's  Physiology  of  Alimentation Large  I2mo,  2  oo 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses i2mo,  2  oo 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  oo 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Wells.)  8vo,  3  oo 

Quantitative  Chemical  Analysis.     (Cohn.)     2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health I2mo,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

*  Getman's  Exercises  in  Physical  Chemistry I2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers I2mo,  i   25 

*  Gooch  and  Browning's  Outlines  of  Qualitative  Chemical  Analysis.  Small  8vo,  i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practfce.     (Woll.) i2mo,  2  oo 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) i2mo,  i  25 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Hanausek's  Microscopy  of  Technical  Products.     (Winton. ) 8vo,  5  oo 

*  Haskin's  and  MacLeod's  Organic  Chemistry 12mo,  2  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,  i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo.  morocco.  2  50 

Herrirk's  Denatured  or  Industrial  Alcohol 8vo,  4  oo 

Hind's  Inorganic  Chemistry 8vo,  3  oo 

*  Laboratory  Manual  for  Students i2mo,  i  oo 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott  )    8vo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,  i  oo 

4 


Holley  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments,  and  Varnishes. 
(In  Press) 

Hopkins's  Oil-chemists'  Handbook ? 8vo,  3  oo 

Iddings's  Rock  Minerals 8vo,   5  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  i  25 
Johannsen's  Key  for  the  Determination  of  Rock -forming  Minerals  in  Thin  Sec- 
tions.    (In  Press) 

Keep's  Cast  Iron T 8vo,  2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis i2mo,  i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

*  Langworthy   and   Austen.         The   Occurrence   of  Aluminium  in   Vegetable 

Products,  Animal'  Products,  and  Natural  Waters 8vo,  2  oo 

Lassar-Cohn's  Application  of  Some  General  Reactions  to  Investigations  in 

Organic  Chemistry.  (Tingle.) i2mo,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Lob's  Electrochemistry  of  Organic  Compounds.  (Lorenz.) 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments 8vo,  3  oo 

Low's  Technical  Method  of  Ore  Analysis 8vo,  3  oo 

Lunge's  Techno-chemical  Analysis.  (Cohn.) 12 mo  i  oo 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Maire's  Modern  Pigments  and  their  vehicles.     (In  Press.) 

Mandel's  Handbook  for  Bio-chemical  Laboratory I2mo,  I  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  i2mo,  60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Matthew's  The  Textile  Fibres.     2d  Edition,  Rewritten      8vo,    400 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .  i2mo, 

Miller's  Manual  of  Assaying 12010, 

Cyanide  Process i2mo, 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) .  .  .  .  i2mo, 

Mixter's  Elementary  Text-book  of  Chemistry I2mo, 

Morgan's  An  Outline  of  the  Theory  of  Solutions  and  its  Results I2mo, 

Elements  of  Physical  Chemistry i2mo,  3  oo 

*  Physical  Chemistry  for  Electrical  Engineers I2mo,  5  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Muir's  History  of  Chemical  Theories  and  Laws 8vo,  4  oo 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo,  150 

"                   "               "           "             Part  Two.     (Turnbull.) 12010,  200 

*  Palmer's  Practical  Test  Book  of  Chemistry I2mo,  1  oo 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.     (Fischer.) .  .  .  .  I2mo,  i  25 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) I2mo,  i  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis I2mo,  i  25 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Standpoint.Svo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Riggs's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 


Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

*  Whys  in  Pharmacy I2mo,  i  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  2  50 

Schimpf  s  Text-book  of  Volumetric  Analysis I2mo,  2  50 

Essentials  of  Volumetric  Analysis I2mo,  i  25 

*  Qualitative  Chemical  Analysis 8vo,  i  25 

Smith's  Lecture  Notes  on  Chemistry  for  Dental  Students 8vo,  2  50 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

Stockbridge's  Rocks  and  Soils 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry 8vo,  3  oo 

Treadwell's  Qualitative  Analysis.     (Hall.) 8vo,  3  oo 

Quantitative  Analysis.     (Hall.) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) i2mo,  i  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining.     Vol.  I Small  8vo,  4  oo 

Vol.11 Small8vo,  500 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Weaver's  Military  Explosives 8vo,  3  oo 

Wehrenfennig's  Analysis  and  Softening  of  Boiler  Feed-Water 8vo,  4  oo 

Wells's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students ^ i2mo,  i  50 

Text-book  of  Chemical  Arithmetic I2mo,  i  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chlorination  Process I2mo,  I  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry - i2mo,  2  oo 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS   OF    ENGINEERING 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mo,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches.  25 

Breed  and  Hosmer's  Principles  and  Practice  of  Surveying ,8vo,  3  oo 

*  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal 8vo,  3  50 

•Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

*  CortheU's  Allowable  Pressures  on  Deep  Foundations I2mo,  i  25 

CrandalTs  Text-book  on  Geodesy  and  Least  Scjuares 8vo,  3  oo 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage i2mo,  i  50 

Practical  Farm  Drainage i2mo,  i  oo 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  oo 

Flemer's  Phototopographic  Methods  and  Instruments 8vo,  5  oo 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements i2mo,  i  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

*  Hauch  and  Rice's  Tables  of  Quantities  for  Preliminary  Estimates I2mo,  i  25 

Hayford's  Text-book  of  Geodetic  Astronomy .8vo,  3  oo 

6 


Bering's  Ready  Reference  Tables  (Conversion  Factors') i6mo,  morocco,  2  50 

Howe's  Retaining  Walls  for  Earth I2mo,  i  25 

Hoyt  and  Grover's  River  Discharge 8vo,  2  oo 

*  Ives's  Adjustments  of  the  Engineer's  Transit  and  Level i6mo,  Bds.  25 

Ives  and  Hilts's  Problems  in  Surveying i6mo,  morocco,  i  50 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.)-  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*  Descriptive  Geometry 8vo»  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying. '. 8vo,  3  50 

Ogden's  Sewer  Design 12 mo,  2  oo 

Parsons's  Disposal  of  Municipal  Refuse 8vo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Riemer's  Shaft-sinking  under  Difficult  Conditions.     (Corning  and  Peele.) .  .8vo,  3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Tracy's  Plane  Surveying I6mo,  morocco,  3  oo 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  oo 

Venable's  Garbage  Crematories  in  America 8vo,  2  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,     i  25 

Wilson's  Topographic  Surveying 8vo,    3  50 

BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations 8vo,  3  oo 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i  25 

Bridge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Grimm's  Secondary  Stresses  in  Bridge  Trusses.     (Tn  Press.) 

Howe's  Treatise  on  Arches 8vo,  4  oo 

Design  of  Simple  Roof -trusses  in  Wood  and  Steel 8vo,  2  oo 

Symmetrical  Masonry  Arches 8vo,  2  50 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.     Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II.     Graphic  Statics 8vo,  2  50 

Part  III.  Bridge  Design 8vo,  2  50 

Part  IV.  Higher  Structures 8vo,  2  50 

7 


Morison's  Memphis  Bridge 4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,     2  oo 

Specifications  for  Steel  Bridges i2mo.  50 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo,    3  50 


HYDRAULICS. 

Barnes's  Ice  Formation 8vo,  3  oo 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i   50 

Hydraulic  Motors 8vo,  2  oo 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12 mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

FrizelFs  Water-power 8vo,  5  oo 

Fuertes's  Water  and  Public  Health i2mo,  i   50 

Water-filtration  Works I2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Bering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Clean  Water  and  How  to  Get  It Large  I2mo,  1  5o 

Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

*  Hubbard  and  Kiersted's  Water- works  Management  and  Maintenance..  .  8vo,  4  co 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo.  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Schuyler's   Reservoirs   for   Irrigation,   Water-power,   and   Domestic   Water- 
supply Large  8vo,  5  oo 

*  Thomas  and  Watt's  Improvement  of  Rivers 4to,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams.     5th  Edition,  enlarged.  .    4to,  6  oo 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 410,  10  oo 

Whipple's  Value  of  Pure  Water Large  i2mo,  i  oo 

Williams  and  Hazen's  Hydraulic  Tables 8vo,  i  50 

Wilson's  Irrigation  Engineering Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements .  8vo,  5  oo 

Black's  United  States  Public  Works Oblong  410,  5  oo 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  5° 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to,  7  50 

*Eckel's  Cements,  Limes,  and  Plas  ers 8vo,  6  oo 

8 


Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

•Graves's  Forest  Mensuration 8vo,  4  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics v  .1 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Maurer's  Technical  Mechanics «*.. 8vo,  4  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mcr.,  4  oo 

*  Ries's  Clays:  Their  Occurrence,  Properties,  and  Uses 8vo,  5  oo 

Rockwell's  Roads  and  Pavements  ih  France I2mo,  i  25 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  ard  Varnish 8vo,  3  oo 

*Schwarz's  Longleaf  Pine  in  Virgin  Forest,., izmo,  i  25 

Smith's  Materials  of  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement 121110,  2  oo 

Text-book  on  Roads  and  Pavements izmo,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  oo 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  oo  ' 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Turneaure  and  Maurer's  Principles  of  Reinforced  Concrete  Construction..  .8vo,  3  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.) .  .  i6mo,  mor.,  2  oo 

*  Specifications  for  Steel  Bridges i2mo,  50 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:  Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo,  4  oo 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  i  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  I  50 

Railway  and  Other  Earthwork  Tables* 8vo,  i  50 

Crockett's  Methods  for  Earthwork  Computations.     (In  Press) 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo.  morocco,  5  oo 

Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) Paper,  5  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 
Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

.Raymond's  Elements  of  Railroad  Engineering.     (In  Press.) 

9 


Searles's  Field  Engineering i6mo  morocco,  3  oo 

Railroad  Spiral. i6mo,  morocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork. ...    ....    8vo,  i  50- 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo>  2  oa 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50- 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo> 

Economics  of  Railroad  Construction .  , Large  i2mo,  2  50 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo 


DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "                   '        Abridged  Ed 8vo.  i  50 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi» 

neers Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oa 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50- 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Advanced  Mechanical  Drawing 8vo,  2  oa 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oa 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;  or,  Practical  Mechanism Svo,  5  oo 

Mechanical  Drawing 4to,  4  oa 

Velocity  Diagrams Svo,  i  50 

MacLeod's  Descriptive  Geometry Small  Svo,  i   50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting Svo,  i   50 

Industrial  Drawing.  (Thompson.) Svo,  3  50 

Moyer's  Descriptive  Geometry Svo,  2  oa 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oa 

Reid's  Course  in  Mechanical  Drawing Svo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. Svo,  3  oo 

Robinson's  Principles  of  Mechanism Svo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism Svo,  3  oo 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.  (McMillan.) Svo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design Svo,  3  oo 

*  Titsworth's  Elements  of  Mechanical  Drawing Oblong  Svo,  25 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo,  oo 


Drafting  Instruments  and  Operations i2mo, 

Manual  of  Elementary  Projection  Drawing i2mo. 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 


oo 


Shadow  ...............................................  i2mo, 

Plane  Problems  in  Elementary  Geometry  ......................  i2mo, 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective  .......  Svo,    3  50 

General  Problems  of  Shades  and  Shadows  .......................  Svo,    3  oo 

Elements  of  Machine  Construction  and  Drawing  ..................  Svo,     7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry  .......  Svo,     2  50 

Weisbach's    Kinematics    and    Power    of    Transmission.        (Hermann    and 

Klein.)  .................................................  Svo,    5  oa 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving  .......  12  mo,    2  oo 

Wilson's  (H.  M.)  Topographic  Surveying  .............................  Svo,    3  50* 

10 


Wilson's  (V.  T.)  Free-hand  Perspective 8vo,    2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,     i  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,    3  oo 

ELECTRICITY  AND  PHYSICS. 

*  Abegg's  Theory  of  Electrolytic  Dissociation.     (Von  Ende.) i2mo,  i  25 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory'of  Electrical  Measurements.  . .  .i2mo,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Betts's  Lead  Refining  and  Electrolysis.     (In  Press.) 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).Svo,  3  oo 

*  Collins's  Manual  of  Wireless  Telegraphy i2mo,  i  50 

Morocco,  2  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

*  Danneel's  Electrochemistry.     (Merriam.) I2mo,  i  25 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's  Theory  of  the  Lead  Accumulator  (Storage  Battery).    (Von  Ende.) 

i2mo, 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo, 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo, 

Gilbert's  De  Magnete.     (Mottelay. ) 8vo, 

Hanchett's  Alternating  Currents  Explained I2mo, 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco, 

Hobart  and  Ellis 's  High-speed  Dynamo  Electric  Machinery.     (In  Press.) 

Holman's  Precision  of  Measurements 8vo,    2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and  Tests.  .  .  .Large  8vo,         75 
Karapetoff's  Experimental  Electrical  Engineering.     (In  Press.) 

Kinzbrunner's  Testing  of  Continuous-current  Machines 8vo,    2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,    3  oo 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,    3  oo 
Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo,    3  oo 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each,    6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,    4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,    2  50 

Norris's  Introduction  to  the  Study  of  Electrical  Engineering.     (In  Press.) 

*  Parshall  and  Hobart's  Electric  Machine  Design 4to,  half  morocco,  12  50 

Reagan's  Locomotives:    Simple,  Compound,  and  Electric.      New  Edition. 

Large  12 mo,  3  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner. ).  .  .8vo,  2  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  2  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,    2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,    7  oo 

*  Sheep,    7  So 

*  Dudley's  Military  Law  and  the  Procedure  of  Courts-martial  ....  Large  i2mo,    2  50 

Manual  for  Courts-martial i6mo,  morocco,    i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo»  5  oo 

Sheep.  5  50 

Law  of  Contracts 8vo.  3  oo 

Winthrop's  Abridgment  of  Military  Law. .  - lamo,  a  50 

11 


MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule 12010,  2  50 

Bolland's  Iron  Founder i2mo,  2  50 

The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding lamo,  3  oo 

*  Claassen's  Beet-sugar  Manufacture.    (Hall  and  Rolfe.) 8vo,  3  oo 

*  Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Eff rent's  Enzymes  and  their  Applications.     (Prescott.) 8vo.  3  oo 

Fitzgerald's  Boston  Machinist i2mo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Herrick's  Denatured  or  Industrial  Alcohol 8vo,  4  oo 

HoUey  and  Ladd's  Analysis  of  Mixed  Paints,  Color  Pigments,  and  Varnishes. 

(In  Press.) 

Hopkins 's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i   50 

Maire's  Modern  Pigments  and  their  Vehicles.     (In  Press.) 

Matthews's  The  Textile  Fibres.     2d  Edition,  Rewritten 8vo,  4  oo 

Metcalf's  Steel.     A  Maunal  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures— And  the  Administration  of  Workshops .   8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo' 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Rice's  Concrete-block  Manufacture 8vo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Press-working  of  Metals 8vo,  3  oo 

Spalding's  Hydraulic  Cement 1 2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

-Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion   8vo,  5  oo 

Ware's  Beet-sugar  Manufacture  and  Refining.     Vol.  I Small  8vo,  4  oo 

Vol.  II 8vo,  5  oo 

Weaver's  Military  Explosives 8vo,  3  oo 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book I2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover '. 8vo,  3  oo 

Wood's  Rustless  Coatings:  Corrosion  and  Electrolysis  ot  Iron  and  Steel .   8vo,  400 

MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

Briggs's  Elements  of  Plane  Analytic  Geometry I2mo,  i  oo 

Buchanan's  Plane  and  Spherical  Trigonometry.     (In  Press.) 

Compton's  Manual  of  Logarithmic  Computations 1 2010,  i   50 

Da  vis's  Introduction  to  the  Logic  of  Algebra 8vo,  i  50 

*  Dickson's  College  Algebra Large  i2mo,  i  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo,  i   25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Halsted's  Elements  of  Geometry 8vo,  i   75 

Elementary  Synthetic  Geometry 8vo,  i   50 

*  Rational  Geometry 12010,  i   50 

12 


*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:  Vest-pocket  size. paper,         15 

100  copies  for     5  oo 

*  Mounted  on  heavy  cardboard,  8  X 10  inches,         25 

10  copies  for     2  oo 
Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus.  .Small  8vo,     3  oo 

Elementary  Treatise  on  the  Integral  Calculus Small  8vo,     i  50 

Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,     3  50 

Johnson's  Treatise  on  the  Integral  Calculus Small  8vo,     3  oo 

Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  121110,     i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,     3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.  ).i2mo,     2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,     3  oo 

Trigonometry  and  Tables  published  separately Each,     2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,     i  oo 

Manning's  IrrationalNumbers  and  their  Representation  bySequences  and  Series 

i2mo,     i  25 
Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward „ Octavo,  each     i  oo 

No.  i.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  <5.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors,  « 

by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
by  Mansfield  Merriman.  No.  n.  Functions  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics 8vo,    4  oo 

Merriman's  Method  of  Least  Squares 8vo,    2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,    2  50 

*  Veblen  and  Lennes's  Introduction  to  the  Real  Infinitesimal  Analysis  of  One 

Variable 8vo,    2  oo 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,    2  oo 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,     i  oo 


MECHANICAL  ENGINEERING. 

MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  50 

Baldwin's  Steam  Heating  for  Buildings 121110,  50 

Barr's  Kinematics  of  Machinery 8vo,  50 

*  Bartlett's  Mechanical  Drawing 8vo,  oo 

*  "                   "                 "        Abridged  Ed 8yo,  50 

Benjamin's  Wrinkles  and  Recipes i2mo,  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings 8vo,  4  oo 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  o*» 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

13 


Durley's  Kinematics  of  Machines 8vo,  4  OO 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  <x> 

Rope  Driving I2mo,  2  oo> 

Gill's  Gas  and  Fuel  Analysis  for  Engineers I2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.)  .  .  8vo,  4  oo 
MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacFarland's  Standard  Reduction  Factors  for  Gases 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  I  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design. 8vo,  3  oo 

Thurston's   Treatise    on    Friction  and   Lost   Work   in   Machinery   and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  i2mo,  I  oo 

Tillson's  Complete  Automobile  Instructor i6mo,  i  50 

Morocco,  2  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — • 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

MATERIALS   OF   ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.     6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Strength  of  Materials i2mo,  i   oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines I2mo,  i   oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

14 


Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,    2  oo 

Elements  of  Analytical  Mechanics 8vo,    3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel . . . 8vo,    4  oo 

STEAM-ENGIlteS  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram izmo,    i  25 

•Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,     i  50 

Creighton's  Steam-engine  and  other  Heat-motors 8vo,    5  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book. . .  .i6mo,  mor.,    5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,    i  oo 

boss's  Locomotive  Sparks 8vo,    2  oo 

Locomotive  Performance 8vo,   5  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2ino,    2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,    5  oo 

Heat  and  Heat-engines 8vo,    5  oo 

Xent's  Steam  boiler  Economy 8vo,    4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,     i  50 

MacCord's  Slide-valves 8vo,    2  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Peabody's  Manual  of  the  Steam-engine  Indicator. I2mo,    i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors 8vo,    i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,    5  oo 

Valve-gears  for  Steam-engines 8vo,    2  50 

Peabody  and  Miller's  Steam-boilers 8vo,    4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,    2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,    i  25 

JReagan's  Locomotives:    Simple,  Compound,  and  Electric.     New  Edition. 

Large  i2mo,    3  50 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,    2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,    2  50 

Snow's  Steam-boiler  Practice 8vo,    3  oo 

Spangler's  Valve-gears 8vo,    2  50 

Notes  on  Thermodynamics i2mo,    i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,    3  oo 

Thomas's  Steam-turbines 8vo,    3  50 

Thurston's  Haudy  Tables 8vo,    i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory 8vo,    6  oo 

Part  II.     Design,  Construction,  and  Operation 8vo,    6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,    5  oo 

Stationary  Steam-engines 8vo,    2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice i2mo,    i  50 

Manual  of  Stpam-boilers,  their  Designs,  Construction,  and  Operation. 8vo,    5  oo 
Wehrenfenning's  Analysis  and  Softening  of  Boiler  Feed-water  (Patterson)   8vo,     4  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,    5  oo 

Whitham's  Steam-engine  Design 8vo,    5  oo 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo,    4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures  8vo,  7  50 

Chase's  The  Art  of  Pattern-making I2tno,  a  50 

15 


Church's  Mechanics  of  Engineering 8vo,  6  oo 

Notes  and  Examples  in  Mechanics 8vo,  2  oo 

Compton's  First  Lessons  in  Metal- working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo,  i  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  i  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,  i  50 

Dingey's  Machine/y  Pattern  Making i2mo,  2  oo 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the   World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics : 

Vol.      I.     Kinematics 8vo,  3  50 

VoL    II.     Statics 8vo,  4  oo 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12 mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Locomotive  Performance 8vo,  5  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  i  oo 

Hobart  and  Ellis 's  High-speed  Dynamo  Electric  Machinery.     (In  Press.) 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinematics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission. 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

"  Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.). 8vo,  4  oo 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

*  Martin's  Text  Book  on  Mechanics,  Vol.  I,  Statics i2mo,  i   25 

*  Vol.  2,  Kinematics  and  Kinetics  .  .I2mo,  l  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

*  Elements  of  Mechanics i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics.  .  . 8vo,  4  oo 

*  Parshall  and  Hobart's  Electric  Machine  Design 4to,  half  morocco,  12  50 

Reagan's  Locomotives :  Simple,  Compound,  and  Electric.     New  Edition. 

Large  12 mo,  3  5o 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Sanborn's  Mechanics :  Problems Large  i2mo,  i   50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Sinclair's  Locomotive-engine  Running  and  Management I2mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Sorer  s    Carbureting  and   Combustion  of   Alcohol  Engines.     (Woodward  and 

Preston.) • Large  8vo,  3  oo 

16 


Spangler  Greene  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work.    ....    8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  i2mo,  i  oo 

Tillson's  Complete  Automobile  Instructor i6mo,  i  50 

Morocco,  2  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing JJvo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein. ) .  8vo,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics .* I2mo,  i  25 

Turbines, .  * .- ' 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

MEDICAL. 

*  Bolduan's  Immune  Sera I2mo,     1  50 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.) Large  i2mo,    2  50 

Ehrlich's  Collected  Studies  on  Immunity.     (Bolduan.) 8vo,    6  oo 

*  Fischer's  Physiology  of  Alimentation Large  12mo,  cloth,     2  oo 

Hammarsten's  Text-book  on  Physiological  Chemistry.     (Mandel.) 8vo,   4  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz.) i2mo, 

*  Pauli's  Physical  Chemistry  in  the  Service  of  Medicine.     (Fischer.) .  .  .  .  I2mo, 

*  Pozzi-Escot's  The  Toxins  and  Venoms  and  their  Antibodies.     (Conn.).  i2mo, 

Rostoski's  Serum  Diagnosis.     (Bolduan.) I2mo, 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) ....  .8vo, 

*  Satterlee's  Outlines  of  Human  Embryology i2mo, 

Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo, 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo, 

Woodhull's  Notes  on  Military  Hygiene i6mo, 

*  Personal  Hygiene i2mo, 

Wulling's  An  Elementary  Course  in  Inorganic  Pharmaceutical  and  Medical 

Chemistry i2mo,    2  oo 

METALLURGY. 

Betts's  Lead  Refining  by  Electrolysis.    (In  Press.) 
Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and  Mercury. , . . , 8vo,  7  50 

Goesel's  Minerals  and  Metals:     A  Reference  Book i6mo,  mor.  3  oo 

*  Iles's  Lead-smelting i2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)i2mo,  3  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users ; i2mo,  2  oo 

Miller's  Cyanide  Process • i2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). ...  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Smith's  Materials  of  Machines I2mo,  I  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  oo 

Part    II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

MINERALOGY. 
Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,    2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,    3  oo 

17 


00 

25 

00 
00 

50 

25 

50 

oo 

50 

00 


Boyd's  Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  150 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  ,8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them I2mo,  i  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography ; i2mo  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Goesel's  Minerals  and  Metals :     A  Reference  Book i6mo,mor.  300 

Groth's  Introduction  to  Chemical  Crystallography  (Marshall) i2mo,  i  25 

Iddings's  Rock  Minerals 8vo,  5  oo 

Johannsen's  Key  for  the  Determination  of  Rock-forming  Minerals    in   Thin 
Sections.    (In  Press.) 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe.  lamo,  60 
Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses 8vo,  4  oo 

Stones  for  Building  and  Decoration ...8vo,  500 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Tables  of  Minerals. , 8vo,  i  00 

*  Richards's  Synopsis  of  Mineral  Characters i2mo,  morocco,  i  25 

*  Ries's  Clays:  Their  Occurrence,  Properties,  and  Uses 8vo,  5  oo 

Rosenbusch's   Microscopical   Physiography   of   the   Rock-making  Minerals. 

(Iddings.) 8vo,  5  oo 

*  Tollman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  oo 

MINING. 

Beard's  Mine  Gases  and  Explosions.     (In  Press.) 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  i  oo 

Eissler's  Modern  High  Explosives 8vo.  4  oo 

Goesel's  Minerals  and  Metals :     A  Reference  Book .  . i6mo,  mor.  3  oo 

Goodyear's  Coal-mines  of  the  Western  Coait  of  the  United  States i2mo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  5  oo 

*  Iles's  Lead-smelting i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Miller's  Cyanide  Process i2mo,  i  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Weaver's  Military  Explosives 8vo,  3  oo 

Wilson's  Cyanide  Processes i2mo,  i  50 

Chlorination  Process isrno,  i  50 

Hydraulic  and  Placer  Mining.     2d  edition,  rewritten 121110,  2  50 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation I2mo,  i  25 

SANITARY  SCIENCE.  . 

Bashore's  Sanitation  of  a  Country  House I2mo,  i  oo 

*  Outlines  of  Practical  Sanitation I2mo,  i  25 

FolwelFs  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oo 

Water-supply  Engineering 8vo,  4  oo 

18 


Fowler's  Sewage  Works  Analyses , I2mo,  oo 

Fuertes's  Water  and  Public  Health I2mo,  50 

Water-filtration  Works. 12010,  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo.  oo 

Sanitation  of  Public  Buildings. . .    . .  t 12mo.  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control „ **: 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint  )8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

*  Merriman's  Elements  of  Sanitary  Engineering , 8vo,  2  oo 

Ogden's  Sewer  Design.  .  . . ,. i2mo.  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis I2mo,  i  25 

*  Price's  Handbook  on  Sanitation i2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Shelter i2mo,  i  oo 

Richards  and  Woodman's  Air.  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  S  wage  and  Bacterial  Purification  of  Sewage 8vo,  4  oo 

Disinfection  and  the  Preservation  of  Food 8vo,  400 

Turneaure  and  Russell's  Public  Water-supplies .8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) 12 mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Air  Conditioning.     (In  Press.) 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

*  Personal  Hygiene , tamo,  i  oo 


MISCELLANEOUS. 

Association  of   State   and  National  Food  and  Dairy  Departments  (Interstate 
Pure  Food  Commission) ; 

Tenth  Annual  Convention  Held  at  Hartford,  July  17-20,  1906.  ...8vo,     3  oo 
Eleventh    Annual    Convention,    Held  at  Jamestown   Tri-Centennial 

Exposition,  July  16-19,  1907.     (In  Press.) 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists. Large  8vo,    I  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo,     4  oo 

Gannett's  Statistical  Abstract  of  the  World 24mo,        75 

Gerhard's  The  Modern  Bath  and  Bath-houses.     (In  Press.) 

Haines's  American  Railway  Management 12 mo,    2  50 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1 824-1 894.. Small  8vo,    3  oo 

Rotherham's  Emphasized  New  Testament.    . Large  8vo,    2  oo 

Standage's  Decorative  Treatment  of  Wood,  Glass,  Metal,  etc.     (In  Press.) 

The  World's  Columbian  !  xposition  of  1893     4to,     i  oc 

Winslow's  Elements  of  Applied  Microscopy .  12 mo,     i  50 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar I2mo,  i  25 

Hebrew  Chrestomathy 8vo.  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  s  oo 

Letteris's  Hebrew  Bible ,...., 8vo,  2  25 

19 


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EEKKEfLJ-V.  CALIFORNIA 


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